SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 1.5 - Vectors and Three Dimensional Geometry [Latest edition]

If |a̅| = 3, |b̅| =4, then the value of λ for which a̅ + λ b̅ is perpendicular to a̅ − λ b̅ is ______ 

`(hat"i" + hat"j" - hat"k")*(hat"i" - hat"j" + hat"k")` = ______.

The angle θ between two non-zero vectors `bar("a")` and `bar("b")` is given by cos θ = ______

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

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

Select the correct option from the given alternatives:

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

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 ______ 

If `|bar("a")|` = 2, `|bar("b")|` = 5, and `bar("a")*bar("b")` = 8 then `|bar("a") - bar("b")|` = ______ 

If `bar("AB") = 2hat"i" + hat"j" - 3hat"k"`, and A(1, 2 ,−1) is given point then coordinates of B are ______.

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 ______ 

The values of c that satisfy `|"c"  bar("u")|` = 3, `bar("u") = hat"i" + 2hat"j" + 3hat"k"` is ______ 

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

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

Find the magnitude of a vector with initial point : (1, −3, 4); terminal point : (1, 0, −1)

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.

A(2, 3), B(−1, 5), C(−1, 1) and D(−7, 5) are four points in the Cartesian plane, Check if, `bar("CD")` is parallel to `bar("AB")`

Find a unit vector in the opposite direction of `baru`. Where `baru = 8hati + 3hatj- hatk`

The non zero vectors `bar("a")` and `bar("b")` are not collinear find the value of `lambda` and `mu`: if `bar("a") + 3bar("b") = 2lambdabar("a") - mubar("b")`

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

Find the distance from (4, −2, 6) to the XZ- plane

If the vectors `2hat"i" - "q"hat"j" + 3hat"k"` and `4hat"i" - 5hat"j" + 6hat"k"` are collinear then find the value of q

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", bar("c") = -5hat"i" + 2hat"j" + 3hat"k"`

If a line makes angle 90°, 60° and 30° with the positive direction of X, Y and Z axes respectively, find its direction cosines

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"|`.

Show that the following points are collinear:

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

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

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

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")`

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

If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.

If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`

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
(i) internally
(ii) externally

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

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.

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

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

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Using properties of scalar triple product, prove that `[(bara + barb,  barb + barc,  barc + bara)] = 2[(bara, barb, barc)]`.

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

If G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

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

Prove that medians of a triangle are concurrent

Prove that altitudes of a triangle are concurrent

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"`

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

Prove that the angle bisectors of a triangle are concurrent

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

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

Let `A(bara)` and `B(barb)` are any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `barr = (mbarb + nbara)/(m + n)`.

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.

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"w" xx bar"r") xx (bar"r" xx bar"w")]`

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).

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")]`.