Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 - Vectors [Latest edition]

Chapter 5: Vectors

Exercise 5.1Exercise 5.2Exercis 5.2Exercise 5.3Exercise 5.4Exercise 5.5Miscellaneous exercise 5
Exercise 5.1 [Pages 151 - 152]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.1 [Pages 151 - 152]

Exercise 5.1 | Q 1 | Page 151

The vector bar"a" is directed due north and |bar"a"| = 24. The vector bar"b" is directed due west and |bar"b"| = 7. Find |bar"a" + bar"b"|.

Exercise 5.1 | Q 2 | Page 151

In the triangle PQR, bar"PQ" = bar"2a", bar"QR" = bar"2b". The midpoint of PR is M. Find the following vectors in terms of bar"a" and bar"b":

(i) bar"PR" (ii) bar"PM" (iii) bar"QM".

Exercise 5.1 | Q 3 | Page 151

OABCDE is a regular hexagon. The points A and B have position vectors bar"a" and bar"b" respectively referred to the origin O. Find, in terms of bar"a" and bar"b" the position vectors of C, D and E.

Exercise 5.1 | Q 4 | Page 151

ABCDEF is a regular hexagon. Show that bar"AB" + bar"AC" + bar"AD" + bar"AE" + bar"AF" = 6bar"AO", where O is the centre of the hexagon.

Exercise 5.1 | Q 5 | Page 151

Check whether the vectors 2hat"i" + 2hat"j" + 3hat"k",  - 3hat"i" + 3hat"j" + 2hat"k" and 3hat"i" + 4hat"k" form a triangle or not.

Exercise 5.1 | Q 6 | Page 151

In the given figure express bar"c" and bar"d" in terms of bar"a" and bar"b".

Exercise 5.1 | Q 7 | Page 151

Find a vector the direction of bar"a" = hat"i" - 2hat"j" that has magnitude 7 units.

Exercise 5.1 | Q 8 | Page 151

Find the distance of (4, - 2, 6) from each of the following:
(a) The XY-plane
(b) The YZ-plane
(c) The XZ-plane
(d) The X-axis
(e) The Y-axis
(f) The Z-axis.

Exercise 5.1 | Q 9.1 | Page 152

Find the coordinates of the point which is located three units behind the YZ-plane, four units to the right of XZ-plane, and five units above the XY-plane.

Exercise 5.1 | Q 9.2 | Page 152

Find the coordinates of the point which is located in the YZ-plane, one unit to the right of the XZ- plane, and six units above the XY-plane.

Exercise 5.1 | Q 10 | Page 152

Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).

Exercise 5.1 | Q 11 | Page 152

If bar"AB" = 2hat"i" - 4hat"j" + 7hat"k" and initial point A(1, 5, 0). Find the terminal point B.

Exercise 5.1 | Q 12.1 | Page 152

Show that the following points are collinear:

A = (3, 2, -4), B = (9, 8, -10), C = (-2, -3, 1)

Exercise 5.1 | Q 12.2 | Page 152

Show that the following points are collinear:

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).

Exercise 5.1 | Q 13 | Page 152

If the vectors 2hat"i" - "q"hat"j" + 3hat"k" and 4hat"i" - 5hat"j" + 6hat"k" are collinear, find q.

Exercise 5.1 | Q 14 | Page 152

Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer.

Exercise 5.1 | Q 15 | Page 152

Express - hat"i" - 3hat"j" + 4hat"k" as the linear combination of the vectors 2hat"i" + hat"j" - 4hat"k", 2hat"i" - hat"j" + 3hat"k" and 3hat"i" + hat"j" - 2hat"k"

Exercise 5.2, Exercis 5.2 [Page 160]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.2, Exercis 5.2 [Page 160]

Exercise 5.2 | Q 1.1 | Page 160

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2hat"i" - hat"j" + 3hat"k"  and - 5hat"i" + 2hat"j" - 5hat"k" in the ratio 3 : 2 is internally.

Exercise 5.2 | Q 1.2 | Page 160

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2hat"i" - hat"j" + 3hat"k" and  - 5hat"i" + 2hat"j" - 5hat"k" in the ratio 3 : 2 is externally.

Exercise 5.2 | Q 2 | Page 160

Find the position vector of midpoint M joining the points L(7, - 6, 12) and N(5, 4, - 2).

Exercise 5.2 | Q 3 | Page 160

If the points A (3, 0, p), B (- 1, q, 3) and C (- 3, 3, 0) are collinear, then find
(i) the ratio in which the point C divides the line segment AB
(ii) the values of p and q.

Exercise 5.2 | Q 4 | Page 160

The position vector of points A and B are 6bar"a" + 2bar"b" and bar"a" - 3bar"b". If the point C divides AB in the ratio 3 : 2, show that the position vector of C is 3bar"a" - bar"b".

Exercise 5.2 | Q 5 | Page 160

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Exercis 5.2 | Q 6 | Page 160

D and E divides sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE

Exercise 5.2 | Q 7 | Page 160

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

Exercise 5.2 | Q 8 | Page 160

Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of the lengths of the parallel sides.

Exercise 5.2 | Q 9 | Page 160

If two of the vertices of a triangle are A (3, 1, 4) and B(− 4, 5, −3) and the centroid of the triangle is at G (−1, 2, 1), then find the coordinates of the third vertex C of the triangle

Exercise 5.2 | Q 10 | Page 160

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Exercise 5.2 | Q 11 | Page 160

If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.

Exercise 5.2 | Q 12 | Page 160

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Exercise 5.3 [Pages 169 - 170]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.3 [Pages 169 - 170]

Exercise 5.3 | Q 1 | Page 169

Find two unit vectors each of which is perpendicular to both bar"u" and bar"v" where bar"u" = 2hat"i" + hat"j" - 2hat"k",  bar"v" = hat"i" + 2hat"j" - 2hat"k".

Exercise 5.3 | Q 2 | Page 169

If bar"a" and bar"b" are two vectors perpendicular to each other, prove that (bar"a" + bar"b")^2 = (bar"a" - bar"b")^2

Exercise 5.3 | Q 3 | Page 169

Find the values of c so that for all real x, the vectors "xc"hat"i" - 6hat"j" + 3hat"k" and "x"hat"i" + 2hat"j" + 2"cx"hat"k" make an obtuse angle.

Exercise 5.3 | Q 4 | Page 169

Show that the sum of the length of projections of "p"hat"i" + "q"hat"j" + "r"hat"k" on the coordinate axes, where p = 2, q = 3 and r = 4 is 9.

Exercise 5.3 | Q 5 | Page 169

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

Exercise 5.3 | Q 6.1 | Page 169

Determine where bar"a" and bar"b" are orthogonal, parallel or neithe:

bar"a" = - 9hat"i" + 6hat"j" + 15hat"k" , bar"b" = 6hat"i" - 4hat"j" - 10hat"k"

Exercise 5.3 | Q 6.2 | Page 169

Determine where bar"a" and bar"b" are orthogonal, parallel or neithe:

bar"a" = 2hat"i" + 3hat"j" - hat"k" , bar"b" = 5hat"i" - 2hat"j" + 4hat"k"

Exercise 5.3 | Q 6.3 | Page 169

Determine where bar"a" and bar"b" are orthogonal, parallel or neithe:

bar"a" = -3/5hat"i" + 1/2hat"j" + 1/3hat"k" , bar"b" = 5hat"i" + 4hat"j" + 3hat"k"

Exercise 5.3 | Q 6.4 | Page 169

Determine where bar"a" and bar"b" are orthogonal, parallel or neithe:

bar"a" = 4hat"i" - hat"j" + 6hat"k" , bar"b" = 5hat"i" - 2hat"j" + 4hat"k"

Exercise 5.3 | Q 7 | Page 169

Find the angle P of the triangle whose vertices are P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1).

Exercise 5.3 | Q 8.1 | Page 169

If bar"p", bar"q" and bar"r" are unit vectors, find bar"p".bar"q".

Exercise 5.3 | Q 8.2 | Page 169

If bar"p", bar"q" and bar"r" are unit vectors, find bar"p".bar"r".

Exercise 5.3 | Q 9 | Page 169

Prove by vector method, that the angle subtended on semicircle is a right angle.

Exercise 5.3 | Q 10 | Page 169

If a vector has direction angles 45° and 60°, find the third direction angle.

Exercise 5.3 | Q 11 | Page 169

If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.

Exercise 5.3 | Q 12 | Page 170

If a line has the direction ratios 4, −12, 18, then find its direction cosines

Exercise 5.3 | Q 13 | Page 170

The direction ratios of bar"AB" are - 2, 2, 1. If A ≡ (4, 1, 5) and l(AB) = 6 units, find B.

Exercise 5.3 | Q 14 | Page 170

Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn - 2nl + 6lm = 0.

Exercise 5.4 [Pages 178 - 179]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.4 [Pages 178 - 179]

Exercise 5.4 | Q 1 | Page 178

If bar"a" = 2hat"i" + 3hat"j" - hat"k", bar"b" = hat"i" - 4hat"j" + 2hat"k", find (bar"a" xx bar"b") xx (bar"a" - bar"b")

Exercise 5.4 | Q 2 | Page 178

Find a unit vector perpendicular to the vectors hat"j" + 2hat"k"  and  hat"i" + hat"j".

Exercise 5.4 | Q 3 | Page 178

If bar"a".bar"b" = sqrt3 and bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k", find the angle between bar"a" and bar"b".

Exercise 5.4 | Q 4 | Page 178

If bar"a" = 2hat"i" + hat"j" - 3hat"k" and  bar"b" = hat"i" - 2hat"j" + hat"k", find a vector of magnitude 5 perpendicular to both bar"a" and bar"b".

Exercise 5.4 | Q 5.1 | Page 178

Find bar"u".bar"v" if |bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8

Exercise 5.4 | Q 5.2 | Page 178

Find |bar"u" xx bar"v"| if |bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12

Exercise 5.4 | Q 6 | Page 178

Prove that 2(bar"a" - bar"b") xx 2(bar"a" + bar"b") = 8(bar"a" xx bar"b")

Exercise 5.4 | Q 7 | Page 178

If bar"a" = hat"i" - 2hat"j" + 3hat"k"  , bar"b" = 4hat"i" - 3hat"j" + hat"k" , bar"c" = hat"i" - hat"j" + 2hat"k" verify that bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"

Exercise 5.4 | Q 8 | Page 178

Find the area of the parallelogram whose adjacent sides are bar"a" = 2hat"i" - 2hat"j" + hat"k" and bar"b" = hat"i" - 3hat"j" - 3hat"k"

Exercise 5.4 | Q 9 | Page 178

Show that vector area of a parallelogram ABCD is 1/2 (bar"AC" xx bar"BD") where AC and BD are its diagonals.

Exercise 5.4 | Q 10 | Page 179

Find the area of parallelogram whose diagonals are determined by the vectors bar"a" = 3hat"i" - hat"j" - 2hat"k" and bar"b" = - hat"i" + 3hat"j" - 3hat"k".

Exercise 5.4 | Q 11 | Page 179

If bar"a", bar"b", bar"c", bar"d" are four distinct vectors such that bar"a" xx bar"b" = bar"c" xx bar"d" and bar"a" xx bar"c" = bar"b" xx bar"d" prove that bar"a" - bar"d" is parallel to bar"b" - bar"c".

Exercise 5.4 | Q 12 | Page 179

If bar"a" = hat"i" + hat"j" + hat"k"  "and"  bar"c" = hat"j" - hat"k", find bar"a" vector bar"b" satisfying bar"a" xx bar"b" = bar"c"  "and"  bar"a".bar"b" = 3

Exercise 5.4 | Q 13 | Page 179

Find bar"a" if bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"

Exercise 5.4 | Q 14 | Page 179

If |bar"a".bar"b"| = |bar"a" xx bar"b"| and bar"a".bar"b" < 0, then find the angle between bar"a"  "and"  bar"b".

Exercise 5.4 | Q 15 | Page 179

Prove, by vector method, that sin (α + β) = sin α . cos β + cos α . sin β

Exercise 5.4 | Q 16.1 | Page 179

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are - 2, 1, - 1 and - 3, - 4, 1

Exercise 5.4 | Q 17 | Page 179

Prove that the two vectors whose direction cosines are given by relations al  + bm + cn = 0 and fmn  + gnl + hlm = 0 are perpendicular, if "f"/"a" + "g"/"b" + "h"/"c" = 0

Exercise 5.4 | Q 18 | Page 179

If A(1, 2, 3) and B(4, 5, 6) are two points, then find the foot of the perpendicular from the point B to the line joining the origin and the point A.

Exercise 5.5 [Pages 183 - 184]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.5 [Pages 183 - 184]

Exercise 5.5 | Q 1 | Page 183

Find bar"a".(bar"b" xx bar"c") if bar"a" = 3hat"i" - hat"j" + 4hat"k" , bar"b" = 2hat"i" + 3hat"j" - hat"k" and bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"

Exercise 5.5 | Q 2 | Page 183

If the vectors 3hat"i" + 5hat"k", 4hat"i" + 2hat"j" - 3hat"k" and 3hat"i" + hat"j" + 4hat"k"  are the coterminus edges of the parallelopiped, then find the volume of the parallelopiped.

Exercise 5.5 | Q 3 | Page 183

If the vectors - 3hat"i" + 4hat"i" - 2hat"k" , hat"i" + 2hat"k" and hat"i" - "p"hat"j" are coplanar, then find the value of p.

Exercise 5.5 | Q 4.1 | Page 184

Prove that [bar"a"  bar"b" + bar"c"  bar"a" + bar"b" + bar"c"] = 0

Exercise 5.5 | Q 4.2 | Page 184

Prove that (bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a" bar"b" bar"c"].

Exercise 5.5 | Q 5 | Page 184

If bar"c" = 3bar"a" - 2bar"b", then prove that [bar"a" bar"b" bar"c"] = 0

Exercise 5.5 | Q 6 | Page 184

If bar"u" = hat"i" - 2hat"j" + hat"k" , bar"r" = 3hat"i" + hat"k" and bar"w" = hat"j" - hat"k" are given vectors, then find (bar"u" + bar"w").[(bar"u" xx bar"r") xx (bar"r" xx bar"w")]

Exercise 5.5 | Q 7 | Page 184

Find the volume of a tetrahedron whose vertices are A (- 1, 2, 3), B (3, - 2, 1), C (2, 1, 3) and D (- 1, 2, 4).

Exercise 5.5 | Q 8 | Page 184

If bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j" and bar"c" = 2hat"i" + hat"j" + 3hat"k", then verify that bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"

Exercise 5.5 | Q 9 | Page 184

If bar"a" = hat"i" - 2hat"j", bar"b" = hat"i" + 2hat"j" , bar"c" = 2hat"i" + hat"j" - 2hat"k", then find (i) bar"a" xx (bar"b" xx bar"c") (ii) (bar"a" xx bar"b") xx bar"c" Are the results same? Justify.

Exercise 5.5 | Q 10 | Page 184

Show that bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"

Miscellaneous exercise 5 [Pages 187 - 189]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Miscellaneous exercise 5 [Pages 187 - 189]

Miscellaneous exercise 5 | Q 1.01 | Page 187

Select the correct option from the given alternatives:

If |bar"a"| = 2, |bar"b"| = 3, |bar"c"| = 4 then [bar"a" + bar"b"    bar"b" + bar"c"    bar"c" - bar"a"] is equal to

• 24

• - 24

• 0

• 48

Miscellaneous exercise 5 | Q 1.02 | Page 188

Select the correct option from the given alternatives:

If |bar"a"| = 3, |bar"b"| = 4, then the value of λ for which bar"a" + lambdabar"b", is perpendicular to bar"a" - lambdabar"b", is

• 9/16

• 3/4

• 3/2

• 4/3

Miscellaneous exercise 5 | Q 1.03 | Page 188

If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______

• sqrt(2)

• sqrt(3)

• 1

• 2

Miscellaneous exercise 5 | Q 1.04 | Page 188

Select the correct option from the given alternatives:

If |bar"a"| = 3, |bar"b"| = 5, |bar"c"| = 7 and bar"a" + bar"b" + bar"c" = bar0, then the angle between bar"a"  "and"  bar"b" is

• pi/2

• pi/3

• pi/4

• pi/6

Miscellaneous exercise 5 | Q 1.05 | Page 188

Select the correct option from the given alternatives:

The volume of tetrahedron whose vectices are (1,-6,10), (-1, -3, 7), (5, -1, λ) and (7, -4, 7) is 11 cu units, then the value of λ is

• 7

• 2

• 1

• 5

Miscellaneous exercise 5 | Q 1.06 | Page 188

Select the correct option from the given alternatives:

If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______

• 30° or 90°

• 45° or 60°

• 90° or 30°

• 60° or 120°

Miscellaneous exercise 5 | Q 1.07 | Page 188

Select the correct option from the given alternatives:

The distance of the point (3, 4, 5) from the Y-axis is ______

• 3

• 5

• sqrt(34)

• sqrt(41)

Miscellaneous exercise 5 | Q 1.08 | Page 188

Select the correct option from the given alternatives:

The line joining the points (2, 1, 8) and (a, b, c) is parallel to the line whose direction ratios are 6, 2, 3. The value of a, b, c are

• 4, 3, - 5

• 1, 2, (- 13)/2

• 10, 5, -2

• 3, 5, 11

Miscellaneous exercise 5 | Q 1.09 | Page 188

Select the correct option from the given alternatives:

If cos α, cos β, cos γ are the direction cosines of a line, then the value of sin2α + sin2β + sin2γ  is ______

• 1

• 2

• 3

• 4

Miscellaneous exercise 5 | Q 1.1 | Page 188

Select the correct option from the given alternatives:

If l, m, n are direction cosines of a line then "l"hat
"i" + "m"hat"j" + "n"hat"k" is ______

• null vector

• the unit vector along the line

• any vector along the line

• a vector perpendicular to the line

Miscellaneous exercise 5 | Q 1.11 | Page 188

Select the correct option from the given alternatives:

If |bar"a"| = 3 and - 1 ≤ k ≤ 2, then |"k"bar"a"| lies in the interval

• [0, 6]

• [-3, 6]

• [3, 6]

• [1, 2]

Miscellaneous exercise 5 | Q 1.12 | Page 188

Select the correct option from the given alternatives:

Let α, β, γ be distinct real numbers. The points with position vectors alphahat"i" + betahat"j" + gammahat"k",  betahat"i" + gammahat"j" + alphahat"k",   gammahat"i" + alphahat"j" + betahat"k"

• are collinear

• form an equilateral triangle

• form a scalene triangle

• form a right angled triangle

Miscellaneous exercise 5 | Q 1.13 | Page 189

Select the correct option from the given alternatives:

Let bar"p"  "and"  bar"q" be the position vectors of P and Q respectively, with respect to O and |bar"p"| = "p", |bar"q"| = "q". The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular; then

• 9p2 = 4q2

• 4p2 = 9q2

• 9p = 4q

• 4p = 9q

Miscellaneous exercise 5 | Q 1.14 | Page 189

Select the correct option from the given alternatives:

The 2 vectors hat"j" + hat"k" and 3hat"i" - hat"j" + 4hat"k" represents the two sides AB and AC respectively of a Δ ABC. The length of the median through A is

• sqrt34/2

• sqrt48/2

• sqrt18

• of the median through A is

Miscellaneous exercise 5 | Q 1.14 | Page 189

Select the correct option from the given alternatives:

The 2 vectors hat"j" + hat"k" and 3hat"i" - hat"j" + 4hat"k" represents the two sides AB and AC respectively of a ΔABC. The length of the median through A is

• sqrt(34)/2

• sqrt(48)/2

• sqrt(18)

• sqrt34

Miscellaneous exercise 5 | Q 1.15 | Page 189

Select the correct option from the given alternatives:

If bar"a"  "and"  bar"b" are unit vectors, then what is the angle between bar"a" and bar"b" for sqrt3bar"a" - bar"b" to be a unit vector?

• 30°

• 45°

• 60°

• 90°

Miscellaneous exercise 5 | Q 1.16 | Page 189

Select the correct option from the given alternatives:

If θ be the angle between any two vectors bar"a"  "and"  bar"b" then |bar"a" . bar"b"| = |bar"a" xx bar"b"|, when θ is equal to

• 0

• pi/4

• pi/2

• pi

Miscellaneous exercise 5 | Q 1.17 | Page 189

Select the correct option from the given alternatives:

The value of hat"i".(hat"j" xx hat"k") + hat"j".(hat"i" xx hat"k") + hat"k".(hat"i" xx hat"j") is

• 0

• - 1

• 1

• 3

Miscellaneous exercise 5 | Q 1.18 | Page 189

Select the correct option from the given alternatives:

Let a, b, c be distinct non-negative numbers. If the vectors "a"hat"i" + "a"hat"j" + "c"hat"k" , hat"i" + hat"k"  "and"  "c"hat"i" + "c"hat"j" + "b"hat"k" lie in a plane, then c is

• the arithmetic mean of a and b

• the geometric mean of a and b

• the harmonic man of a and b

• 0

Miscellaneous exercise 5 | Q 1.19 | Page 189

Select the correct option from the given alternatives:

Let bar"a" = hat"i" - hat"j", bar"b" = hat"j" - hat"k", bar"c" = hat"k" - hat"i". If bar"d" is a unit vector such that bar"a". bar"d" = 0 = [bar"b" bar"c" bar"d"], then bar"d" equals

• +- (hat"i" + hat"j" - 2hat"k")/sqrt6

• +- (hat"i" + hat"j" + hat"k")/sqrt3

• +-(hat"i" + hat"j" - hat"k")/sqrt3

• +-  hat"k"

Miscellaneous exercise 5 | Q 1.2 | Page 189

Select the correct option from the given alternatives:

If bar"a", bar"b", bar"c" are non-coplanar unit vectors such that bar"a"xx (bar"b"xxbar"c") = (bar"b"+bar"c")/sqrt2, then the angle between bar"a"  "and"  bar"b" is

• (3pi)/4

• pi/4

• pi/2

• pi

Miscellaneous exercise 5 [Pages 190 - 193]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Miscellaneous exercise 5 [Pages 190 - 193]

Miscellaneous exercise 5 | Q 1 | Page 190

ABCD is a trapezium with AB parallel to DC and DC = 3AB. M is the midpoint of DC. bar"AB" = bar"p", bar"BC" = bar"q".
Find in terms of bar"p" and bar"q":

(i) bar"AM" (ii) bar"BD" (iii) bar "MB" (iv) bar"DA"

Miscellaneous exercise 5 | Q 2 | Page 190

The points A, B, C have position vectors bar"a", bar"b" and bar"c" respectively. The point P is the midpoint of AB. Find the vector bar"PC" in terms of bar"a", bar"b", bar"c".

Miscellaneous exercise 5 | Q 3 | Page 190

In a pentagon ABCDE, show that bar"AB" + bar"AE" + bar"BC" + bar"DC" + bar"ED" = 2bar"AC"

Miscellaneous exercise 5 | Q 4 | Page 190

In a parallelogram ABCD, diagonal vectors are bar"AC" = 2hat"i" + 3hat"j" + 4hat"k" and bar"BD" = - 6hat"i" + 7hat"j" - 2hat"k", then find the adjacent side vectors bar"AB" and bar"AD".

Miscellaneous exercise 5 | Q 5 | Page 190

If two sides of a triangle are hat"i" + 2hat"j" and hat"i" + hat"k", find the length of the third side.

Miscellaneous exercise 5 | Q 6 | Page 190

If |bar"a"| = |bar"b"| = 1,  bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find"  |bar"c"|.

Miscellaneous exercise 5 | Q 7.1 | Page 190

Find the lengths of the sides of the triangle and also determine the type of a triangle:

A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)

Miscellaneous exercise 5 | Q 7.2 | Page 190

Find the lengths of the sides of the triangle and also determine the type of a triangle:

L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).

Miscellaneous exercise 5 | Q 8.1 | Page 190

Find the component form of bar"a" if it lies in YZ-plane makes 60° with positive Y-axis and |bar"a"| = 4.

Miscellaneous exercise 5 | Q 9 | Page 190

Two sides of a parallelogram are 3hat"i" + 4hat"j" - 5hat"k" and  -2hat"j" + 7hat"k". Find unit vectors parallel to the diagonals.

Miscellaneous exercise 5 | Q 10 | Page 190

If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that bar"AD" + bar"BE" + bar"CF" = bar0.

Miscellaneous exercise 5 | Q 11 | Page 190

Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).

Miscellaneous exercise 5 | Q 12 | Page 190

Express hat"i" + 4hat"j" - 4hat"k" as the linear combination of the vectors 2hat"i" - hat"j" + 3hat"k", hat"i" - 2hat"j" + 4hat"k" and - hat"i" + 3hat"j" - 5hat"k".

Miscellaneous exercise 5 | Q 13 | Page 190

If bar"OA" = bar"a" and bar"OB" = bar"b", then show that the vector along the angle bisector of ∠AOB is given by bar"d" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|).

Miscellaneous exercise 5 | Q 15 | Page 191

A point P with position vector (- 14hat"i" + 39hat"j" + 28hat"k")/5 divides the line joining A (1, 6, 5) and B in the ratio 3 : 2, then find the point B.

Miscellaneous exercise 5 | Q 16 | Page 191

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Miscellaneous exercise 5 | Q 17 | Page 191

ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.

Miscellaneous exercise 5 | Q 18 | Page 191

If ABC is a triangle whose orthocentre is P and the circumcentre is Q, prove that bar"PA" + bar"PB" + bar"PC" = 2bar"PQ".

Miscellaneous exercise 5 | Q 19 | Page 191

If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that bar"QP" = 3bar"QG".

Miscellaneous exercise 5 | Q 20 | Page 191

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Miscellaneous exercise 5 | Q 21 | Page 191

Dot product of a vector with vectors 3hat"i" - 5hat"k",  2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k" are respectively -1, 6 and 5. Find the vector.

Miscellaneous exercise 5 | Q 22 | Page 191

If bar"a", bar"b", bar"c" are unit vectors such that bar"a" + bar"b" + bar"c" = bar0, then find the value of bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".

Miscellaneous exercise 5 | Q 23 | Page 191

If a parallelogram is constructed on the vectors bar"a" = 3bar"p" - bar"q", bar"b" = bar"p" + 3bar"q" and |bar"p"| = |bar"q"| = 2 and angle between bar"p" and bar"q" is pi/3, and angle between lengths of the sides is sqrt7 : sqrt13.

Miscellaneous exercise 5 | Q 24 | Page 191

Express the vector bar"a" = 5hat"i" - 2hat"j" + 5hat"k" as a sum of two vectors such that one is parallel to the vector bar"b" = 3hat"i" + hat"k" and other is perpendicular to bar"b".

Miscellaneous exercise 5 | Q 25 | Page 191

Find two unit vectors each of which makes equal angles with bar"u", bar"v" and bar"w" where bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v" = hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k".

Miscellaneous exercise 5 | Q 26 | Page 191

Find the acute angle between the curves at their points of intersection, y = x2, y = x3.

Miscellaneous exercise 5 | Q 27.1 | Page 191

Find the direction cosines and direction angles of the vector 2hat"i" + hat"j" + 2hat"k"

Miscellaneous exercise 5 | Q 28 | Page 191

Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively.

Miscellaneous exercise 5 | Q 29 | Page 192

Show that no line in space can make angles pi/6 and pi/4 with X-axis and Y-axis.

Miscellaneous exercise 5 | Q 30 | Page 192

Find the angle between the lines whose direction cosines are given by the equations 6mn - 2nl + 5lm = 0, 3l + m + 5n = 0.

Miscellaneous exercise 5 | Q 31 | Page 192

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Miscellaneous exercise 5 | Q 32 | Page 192

Show that the vector area of a triangle ABC, the position vectors of whose vertices are bar"a", bar"b" and bar"c" is 1/2[bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"].

Miscellaneous exercise 5 | Q 33 | Page 192

Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?

Miscellaneous exercise 5 | Q 34.01 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a".(bar"b" xx bar"c")

Miscellaneous exercise 5 | Q 34.02 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a" xx (bar"b".bar"c")

Miscellaneous exercise 5 | Q 34.03 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a" xx(bar"b" xx bar"c")

Miscellaneous exercise 5 | Q 34.04 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a".(bar"b".bar"c")

Miscellaneous exercise 5 | Q 34.05 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

(bar"a".bar"b") xx (bar"c".bar"d")

Miscellaneous exercise 5 | Q 34.06 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

(bar"a" xx bar"b").(bar"c"xxbar"d")

Miscellaneous exercise 5 | Q 34.07 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

(bar"a".bar"b").bar"c"

Miscellaneous exercise 5 | Q 34.08 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

(bar"a".bar"b")bar"c"

Miscellaneous exercise 5 | Q 34.09 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

|bar"a"|(bar"b".bar"c")

Miscellaneous exercise 5 | Q 34.1 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a".(bar"b" + bar"c")

Miscellaneous exercise 5 | Q 34.11 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

bar"a". bar"b" + bar"c"

Miscellaneous exercise 5 | Q 34.12 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

|bar"a"|. (bar"b" + bar"c")

Miscellaneous exercise 5 | Q 35 | Page 192

For any vectors bar"a", bar"b", bar"c" show that (bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a" = 2bar"a" xx bar"c"

Miscellaneous exercise 5 | Q 36.1 | Page 192

Suppose bar"a" = bar"0":

If bar"a".bar"b" = bar"a".bar"c", then is bar"b" = bar"c" ?

Miscellaneous exercise 5 | Q 36.2 | Page 192

Suppose bar"a" = bar"0":

If bar"a" xx bar"b" = bar"a" xx bar"c", then is bar"b" = bar"c" ?

Miscellaneous exercise 5 | Q 36.3 | Page 192

Suppose bar"a" = bar"0":

If bar"a".bar"b" = bar"a".bar"c" and bar"a" xx bar"b" = bar"a" xx bar"c",  then is bar"b" = bar"c"?

Miscellaneous exercise 5 | Q 37.1 | Page 192

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then verify that the points are the vertices of a parallelogram.

Miscellaneous exercise 5 | Q 37.2 | Page 192

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then find its area.

Miscellaneous exercise 5 | Q 38 | Page 193

Let A, B, C, D be any four points in space. Prove that |bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" + bar"BD"| = 4 (area of triangle ABC).

Miscellaneous exercise 5 | Q 39 | Page 192

Let hat"a", hat"b", hat"c" be unit vectors such that hat"a".hat"b" = hat"a".hat"c" = 0 and 6  the angle between hat"b" and hat"c" is pi/6. Prove that hat"a" = +- 2(hat"b" xx hat"c").

Miscellaneous exercise 5 | Q 40 | Page 192

Find the value of ‘a’ so that the volume of parallelopiped formed by hat"i" + "a"hat"j" + hat"k", hat"j" + "a"hat"k" and "a"hat"i" + hat"k" becomes minimum.

Miscellaneous exercise 5 | Q 41 | Page 193

Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.

Miscellaneous exercise 5 | Q 42 | Page 192

If bar"a", bar"b", bar"c" are three non-coplanar vectors show that (bar"a".(bar"b" xx bar"c"))/((bar"c" xx bar"a").bar"b") + (bar"b".(bar"a" xx bar"c"))/((bar"c" xx bar"a").bar"b") = 0

Miscellaneous exercise 5 | Q 43 | Page 193

Prove that (bar"a" xx bar"b").(bar"c" xx bar"d") =
|bar"a".bar"c"    bar"b".bar"c"|
|bar"a".bar"d"    bar"b".bar"d"|.

Miscellaneous exercise 5 | Q 44 | Page 193

Find the volume of a parallelopiped whose coterimus edges are represented by the vectors hat"i" + hat"k", hat"i" + hat"k", hat"i" + hat"j". Also find volume of tetrahedron having these coterminus edges.

Miscellaneous exercise 5 | Q 45 | Page 193

Using properties of scalar triple product, prove that [(bar"a" + bar"b",  bar"b" + bar"c",  bar"c" + bar"a")] = 2[(bar"a",  bar"b",  bar"c")].

Miscellaneous exercise 5 | Q 46 | Page 193

If four points "A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d") are coplanar, then show that [(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")].

Miscellaneous exercise 5 | Q 47 | Page 193

If bar"a", bar"b", bar"c" are three non-coplanar vectors, then (bar"a" + bar"b" + bar"c").[(bar"a" + bar"b") xx (bar"a" + bar"c")] = - [bar"a"  bar"b" bar"c"]

Miscellaneous exercise 5 | Q 48 | Page 193

If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair is also perpendicular.

Chapter 5: Vectors

Exercise 5.1Exercise 5.2Exercis 5.2Exercise 5.3Exercise 5.4Exercise 5.5Miscellaneous exercise 5

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 - Vectors

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 Vectors are Representation of Vector, Vectors and Their Types, Algebra of Vectors, Coplanar Vectors, Vector in Two Dimensions (2-D), Three Dimensional (3-D) Coordinate System, Components of Vector, Position Vector of a Point P(X, Y, Z) in Space, Component Form of a Position Vector, Vector Joining Two Points, Section Formula, Scalar Product of Vectors (Dot), Vector Product of Vectors (Cross), Scalar Triple Product of Vectors, Vector Triple Product, Addition of Vectors.

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