Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 8 - Vector Algebra [Latest edition]

Represent graphically the displacement of 45 cm, 30° north of east

Represent graphically the displacement of 80 km, 60° south of west

Prove that the relation R defined on the set V of all vectors by `vec"a"  "R"  vec"b"`  if  `vec"a" = vec"b"` is an equivalence relation on V

Let `vec"a"` and `vec"b"` be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are `(vec"a" + 2vec"b")/3` and `(vec"b" + 2vec"a")/3`

If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that `vec"BE" + vec"DC" = 3/2vec"BC"`

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side

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

If `vec"a"` and `vec"b"` represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal

If `vec"PO" + vec"OQ" = vec"QO" + vec"OR"`, prove that the points P, Q, R are collinear

If D is the midpoint of the aide BC of a triangle ABC, prove that `vec"AB" + vec"AC" = 2vec"AD"`

If G is the centroid of a triangle ABC, prove that `vec"GA" + vec"GB" + vec"GC" = vec0`

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that `vec"AD" + vec"BE" + vec"CF" = vec0`

If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then Prove that `vec"AB" + vec"AD" + vec"CB" + vec"CD" = 4vec"EF"`

Verify whether the following ratios are direction cosines of some vector or not

`1/5, 3/5, 4/5`

Verify whether the following ratios are direction cosines of some vector or not

`1/sqrt(2), 1/2, 1/2`

Verify whether the following ratios are direction cosines of some vector or not

`4/3, 0, 3/4`

Find the direction cosines of a vector whose direction ratios are
1, 2, 3

Find the direction cosines of a vector whose direction ratios are

`1/sqrt(2), 1/2, 1/2`

Find the direction cosines of a vector whose direction ratios are
0, 0, 7

Find the direction cosines and direction ratios for the following vector

`3hat"i" - 4hat"j" + 8hat"k"`

Find the direction cosines and direction ratios for the following vector

`3hat"i" + hat"j" + hat"k"`

Find the direction cosines and direction ratios for the following vector

`hat"j"`

Find the direction cosines and direction ratios for the following vector

`5hat"i" - 3hat"j" - 48hat"k"`

Find the direction cosines and direction ratios for the following vector

`3hat"i" - 3hat"k" + 4hat"j"`

Find the direction cosines and direction ratios for the following vector

`hat"i" - hat"k"`

A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians

If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a

If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c

Show that the vectors `- 2hat"i" - hat"j" - hat"k", - 3hat"i" - 4hat"j" - 4hat"k", hat"i" - 3hat"j" - 5hat"k"` form a right angled triangle

Find the value of λ for which the vectors `vec"a" = 3hat"i" + 2hat"j" + 9hat"k"` and `hat"b" = hat"i" + lambdahat"j" + 3hat"k"` are parallel

Show that the following vectors are coplanar

`hat"i" - 2hat"j" + 3hat"k", -2hat"i" + 3hat"j" - 4hat"k", -hat"j" + 2hat"k"`

Show that the following vectors are coplanar

`2hat"i" + 3hat"j" + hat"k", hat"i" - hat"j", 7hat"i" + 3hat"j" + 2hat"k"`

Show that the points whose position vectors `4hat"i" + 5hat"j" - hat"k", - hat"j" - hat"k", 3hat"i" + 9hat"j" + 4hat"k"` and `-4hat"i" + 4hat"j" + 4hat"k"` are coplanar

If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`,  find the magnitude and direction cosines of `vec"a", vec"b", vec"c"`

If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`,  find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`

The position vectors of the vertices of a triangle are `hat"i" + 2hat"j" + 3hat"k", 3hat"i" - 4hat"j" + 5hat"k"` and `-2hat"i" + 3hat"j" - 7hat"k"`. Find the perimeter of the triangle

Find the unit vector parallel to `3vec"a" - 2vec"b" + 4vec"C"` if `vec"a" = 3hat"i" - hat"j" - 4hat"k", vec"b" = -2hat"i" + 4hat"j" - 3hat"k"`, and `vec"c" = hat"i" + 2hat"j" - hat"k"`

The position vectors `vec"a", vec"b", vec"c"` of three points satisfy the relation `2vec"a" - 7vec"b" + 5vec"c" = vec0`. Are these points collinear?

The position vectors of the points P, Q, R, S are `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2hat"j" - 3hat"k"`, and `hat"i" - 6hat"j" - hat"k"` respectively. Prove that the line PQ and RS are parallel

Find the value or values of m for which `"m"(hat"i" + hat"j" + hat"k")` is a unit vector

Show that the points A(1, 1, 1), B(1, 2, 3) and C(2, – 1, 1) are vertices of an isosceles triangle

Find `vec"a"*vec"b"` when `vec"a" = hat"i" - 2hat"j" + hat"k"` and `vec"b" = 3hat"i" - 4hat"j" - 2hat"k"`

Find `vec"a"*vec"b"` when `vec"a" = 2hat"i" + 2hat"j" - hat"k"` and `vec"b" = 6hat"i" - 3hat"j" + 2hat"k"`

Find the value λ for which the vectors `vec"a"` and  `vec"b"` are perpendicular, where `vec"a" = 2hat"i" + lambdahat"j" + hat"k"` and `vec"b" = hat"i" - 2hat"j" + 3hat"k"`

Find the value λ for which the vectors `vec"a"` and `vec"b"` are perpendicular, where `vec"a" = 2hat"i" + 4hat"j" - hat"k"` and `vec"b" = 3hat"i" - 2hat"j" + lambdahat"k"`

If `vec"a"` and `vec"b"` are two vectors such that `|vec"a"| = 10, |vec"b"| = 15` and `vec"a"*vec"b" = 75sqrt(2)`, find the angle between `vec"a"` and `vec"b"`

Find the angle between the vectors

`2hat"i" + 3hat"j" - 6hat"k"` and `6hat"i" - 3hat"j" + 2hat"k"`

Find the angle between the vectors

`hat"i" - hat"j"` and `hat"j" - hat"k"`

If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + 2vec"b" + vec"c"` = 0 and `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 7`, find the angle between `vec"a"` and `vec"b"`

Show that the vectors `vec"a" = 2hat"i" + 3hat"j" + 3hat"j" + 6hat"k", vec"b" = 6hat"i" + 2hat"j" - 3hat"k"` and `vec"c" = 3hat"i" - 6hat"j" + 6hat"k"` are mutually orthogonal

Show that the vectors `-hat"i" - 2hat"j" - 6hat"k", 2hat"i" - hat"j" + hat"k"` and find `-hat"i" + 3hat"j" + 5hat"k"` form a right angled triangle

If `|vec"a"|= 5, |vec"b"| = 6, |vec"c"| = 7` and `vec"a" + vec"b" + vec"c" = vec"0"`, find `vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a"`

Show that the points (2, –1, 3), (4, 3, 1) and (3, 1, 2) are collinear

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`sin  theta/2 = 1/2|vec"a" - vec"b"|`

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`cos  theta/2 = 1/2|vec"a" + vec"b"|`

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`tan  theta/2 = |vec"a" - vec"b"|/|vec"a" + vec"b"|`

Let `vec"a", vec"b", vec"c"` be three vectors such that `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 5` and each one of them being perpendicular to the sum of the other two, find `|vec"a" + vec"b" + vec"c"|`

Find the projection of the vector `hat"i" + 3hat"j" + 7hat"k"` on the vector `2hat"i" + 6hat"j" + 3hat"k"`

Find λ, when the projection of `vec"a" = lambdahat"i" + hat"j" + 4hat"k"` on `vec"b" = 2hat"i" + 6hat"j" + 3hat"k"` is 4 units

Three vectors `vec"a", vec"b"` and `vec"c"` are such that `|vec"a"| = 2, |vec"b"| = 3, |vec"c"| = 4`, and `vec"a" + vec"b" + vec"c" = vec0`. Find `4vec"a"*vec"b" + 3vec"b"*vec"c" + 3vec"c"*vec"a"`

Find the magnitude of `vec"a" xx vec"b"` if `vec"a" = 2hat"i" + hat"j" + 3hat"k"` and `vec"b" = 3hat"i" + 5hat"j" - 2hat"k"`

Show that `vec"a" xx (vec"b" + vec"c") + vec"b" xx (vec"c" + vec"a") + vec"c" xx (vec"a" + vec"b") = vec0`

Find the vectors of magnitude `10sqrt(3)` that are perpendicular to the plane which contains `hat"i" + 2hat"j" + hat"k"` and `hat"i" + 3hat"j" + 4hat"k"`

Find the unit vectors perpendicular to each of the vectors `vec"a" + vec"b"` and `vec"a" - vec"b"`, where `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`

Find the area of the parallelogram whose two adjacent sides are determined by the vectors  `hat"i" + 2hat"j" + 3hat"k"` and `3hat"i" - 2hat"j" + hat"k"`

Find the area of the triangle whose vertices are A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)

If `vec"a", vec"b", vec"c"` are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is `1/2 |vec"a" xx vec"b" + vec"b" xx vec"c" + vec"c" xx vec"a"|`. Also deduce the condition for collinearity of the points A, B, and C

For any vector `vec"a"` prove that `|vec"a" xx hat"i"|^2 + |vec"a" xx hat"j"|^2 + |vec"a" xx hat"k"|^2 = 2|vec"a"|^2`

Let `vec"a", vec"b", vec"c"` be unit vectors such that `vec"a" * vec"b" = vec"a"*vec"c"` = 0 and the angle between `vec"b"` and `vec"c"` is `pi/3`. Prove that `vec"a" = +-  2/sqrt(3) (vec"b" xx vec"c")`

Find the angle between the vectors `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"` using vector product

Choose the correct alternative:
The value of `vec"AB" + vec"BC" + vec"DA" + vec"CD"` is

Choose the correct alternative:
If `vec"a" + 2vec"b"` and `3vec"a" + "m"vec"b"` are parallel, then the value of m is

Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is

Choose the correct alternative:
A vector `vec"OP"` makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between `vec"OP"` and the z-axis is

Choose the correct alternative:
If `vec"BA" = 3hat"i" + 2hat"j" + hat"k"` and the position vector of is `hat"i" + 3hat"j" - hat"k"`, then the position vector A is

Choose the correct alternative:
A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to

Choose the correct alternative:
The vectors `vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"` are

Choose the correct alternative:
If ABCD is a parallelogram, then `vec"AB" + vec"AD" + vec"CB" + vec"CD"` is equal to

Choose the correct alternative:
One of the diagonals of parallelogram ABCD with `vec"a"` and `vec"b"` as adjacent sides is `vec"a" + vec"b"`. The other diagonal `vec"BD"` is

Choose the correct alternative:
If `vec"a", vec"b"` are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

Choose the correct alternative:
If `vec"a", vec"b", vec"c"` are the position vectors of three collinear points, then which of the following is true?

Choose the correct alternative:
If `vec"r" = (9vec"a" + 7vec"b")/16`, then the point P whose position vector `vec"r"` divides the line joining the points with position vectors `vec"a"` and `vec"b"` in the ratio

Choose the correct alternative:
If `lambdahat"i" + 2lambdahat"j" + 2lambdahat"k"` is a unit vector, then the value of `lambda` is

Choose the correct alternative:
Two vertices of a triangle have position vectors `3hat"i" + 4hat"j" - 4hat"k"` and `2hat"i" + 3hat"j" + 4hat"k"`. If the position vector of the centroid is `hat"i" + 2hat"j" + 3hat"k"`, then the position vector of the third vertex is

Choose the correct alternative:
If `|vec"a" + vec"b"| = 60, |vec"a" - vec"b"| = 40` and `|vec"b"| = 46`, then `|vec"a"|` is

Choose the correct alternative:
If `vec"a"` and `vec"b"` having same magnitude and angle between them is 60° and their scalar product `1/2` is then `|vec"a"|` is

Choose the correct alternative:
The value of θ ∈ `(0, pi/2)` for which the vectors `"a" = (sin theta)hat"i" = (cos theta)hat"j"` and `vec"b" = hat"i" - sqrt(3)hat"j" + 2hat"k"` are perpendicular, equaal to

Choose the correct alternative:
If `|vec"a"| = 13, |vec"b"| = 5` and `vec"a" * vec"b"` = 60° then `|vec"a" xx vec"b"|` is  

Choose the correct alternative:
Vectors `vec"a"` and `vec"b"` are inclined at an angle θ = 120°. If `vec"a"| = 1, |vec"b"| = 2`, then `[(vec"a" + 3vec"b") xx (3vec"a" - vec"b")]^2` is equal to

Choose the correct alternative:
If `vec"a"` and `vec"b"` are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between `vec"a"` and `vec"a" + vec"b"` is

Choose the correct alternative:
If the projection of `5hat"i" -  hat"j" - 3hat"k"` on the vector `hat"i" + 3hat"j" + lambdahat"k"` is same as the projection of `hat"i" + 3hat"j" + lambdahat"k"` on `5hat"i" -  hat"j" - 3hat"k"`, then λ is equal to

Choose the correct alternative:
If (1, 2, 4) and (2, – 3λ – 3) are the initial and terminal points of the vector `hat"i" + 5hat"j" - 7hat"k"` then the value of λ is equal to

Choose the correct alternative:
If the points whose position vectors `10hat"i" + 3hat"j", 12hat"i" - 5hat"j"` and `"a"hat"i" + 11hat"j"` are collinear then a is equal to

Choose the correct alternative:
If `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = 2hat"i" + xhat"j" + hat"k", vec"c" = hat"i" - hat"j" + 4hat"k"` and `vec"a" * (vec"b" xx vec"c")` = 70, then x is equal to

Choose the correct alternative:
If `vec"a" = hat"i" + 2hat"j" + 2hat"k", |vec"b"|` = 5 and the angle between `vec"a"` and `vec"b"` is `pi/6`, then the area of the triangle formed by these two vectors as two sides, is