NCERT solutions for Mathematics [English] Class 12 chapter 10 - Vector Algebra [Latest edition]

Classify the following measures as scalar and vector.

10 kg

Classify the following measures as scalar and vector.

2 meters north-west

Classify the following measures as scalar and vector.

40°

Classify the following measures as scalar and vector.

40 watt

Classify the following measures as scalar and vector.

10^-19 coulomb

Classify the following measures as scalar and vector.

20 m/s²

Classify the following as scalar and vector quantity.

Time period

Classify the following as scalar and vector quantity.

Distance

Classify the following as scalar and vector quantity.

Force

Classify the following as scalar and vector quantity.

Velocity

Classify the following as scalar and vector quantity.

Work done

In Figure, identify the following vector.

Coinitial

In Figure, identify the following vector.

Equal

In Figure, identify the following vector.

Collinear but not equal

`veca and -veca` are collinear.

Two collinear vectors are always equal in magnitude.

Two vectors having the same magnitude are collinear.

Two collinear vectors having the same magnitude are equal.

Compute the magnitude of the following vector:

`veca = hati + hatj + hatk;` `vecb = 2hati - 7hatj - 3hatk`;  `vecc = 1/sqrt3 hati + 1/sqrt3 hatj - 1/sqrt3 hatk`

Find the values of x and y so that the vectors `2hati + 3hatj and xhati  + yhatj` are equal.

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Find the sum of the vectors `veca = hati -2hatj + hatk, vecb = -2hati + 4hatj + 5hatk and vecc = hati - 6hatj - 7hatk.`

Find the unit vector in the direction of the vector `veca = hati + hatj + 2hatk`.

Find the unit vector in the direction of vector `vec(PQ)`, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

For given vectors,  `veca = 2hati - hatj + 2hatk` and `vecb = -hati  + hatj - hatk`, find the unit vector in the direction of the vector `veca +vecb`.

Show that the vectors `2hati - 3hatj + 4hatk` and `-4hati + 6hatj -  8hatk` are collinear.

Find the direction cosines of the vector `hati + 2hatj + 3hatk`.

Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  `hati + 2hatj - hatk` and `-hati + hatj + hatk`  respectively, internally the ratio 2:1.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  `hati + 2hatj - hatk` and `-hati + hatj + hatk`  respectively, externally in the ratio 2:1.

Show that the points A, B and C with position vectors `veca = 3hati - 4hatj - 4hatk`, `vecb = 2hati - hatj + hatk` and `vecc = hati - 3hatj - 5hatk`, respectively form the vertices of a right angled triangle.

In triangle ABC, which of the following is not true:

If `veca` and `vecb` are two collinear vectors, then which of the following are incorrect:

Find the angle between the vectors `hati - 2hatj + 3hatk` and `3hati - 2hatj + hatk`.

Find the projection of the vector `hati - hatj` on the vector `hati + hatj`.

Find the projection of the vector `hati + 3hatj + 7hatk`  on the vector `7hati - hatj + 8hatk`.

Show that each of the given three vectors is a unit vector:

`1/7 (2hati + 3hatj + 6hatj), 1/7(3hati - 6hatj + 2hatk), 1/7(6hati + 2hatj - 3hatk)`

Find `|veca| and |vecb|`, if `(veca + vecb).(veca -vecb) = 8 and |veca| = 8|vecb|.`

Evaluate the product `(3veca - 5vecb).(2veca + 7vecb)`.

Find the magnitude of two vectors `veca and vecb`, having the same magnitude and such that the angle between them is 60° and their scalar product is `1/2`.

Find `|vecx|`, if for a unit vector veca , `(vecx -  veca).(vecx + veca) = 12`.

If `veca = 2hati + 2hatj + 3hatk,  vecb = -veci + 2hatj + hatk and vecc = 3hati + hatj` are such that `veca + lambdavecb`  is perpendicular to `vecc`, then find the value of λ.

If  `veca.veca = 0` and `veca . vecb = 0,` then what can be concluded about the vector `vecb`?

If  `vec a, vec b, vec c`  are unit vectors such that `veca+vecb+vecc=0`, then write the value of  `vec a.vecb+vecb.vecc+vecc.vec a`.

If either vector `veca = vec0`  or `vecb = vec0`, then `veca.vecb = 0`. But the converse need not be true. Justify your answer with an example.

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectors `bar(BA)` and `bar(BC)`].

Show that the vectors `2hati - hatj + hatk, hati - 3hatj - 5hatk`  and `3hati - 4hatj - 4hatk` from the vertices of a right angled triangle.

If `veca` is a nonzero vector of magnitude 'a' and λ a nonzero scalar, then λ`veca` is unit vector if ______.

Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.

Find a unit vector perpendicular to each of the vector  `veca  + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj  - 2hatk`.

If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.

Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.

Find λ and μ if  `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.

Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?

Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`

If either `veca = vec0`  or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Let the vectors `veca` and `vecb` be such that `|veca| = 3` and `|vecb| = sqrt2/3`, then `veca xx vecb` is a unit vector, if the angle between `veca` and `vecb` is ______.

Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.

Find the scalar components and magnitude of the vector joining the points `P(x_1, y_1, z_1) and Q (x_2, y_2, z_2).`

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors `veca = 2i + 3hatj - hatk` and `vecb = hati - 2hatj + hatk`.

If `veca = hati  +hatj + hatk, vecb = 2hati - hatj +  3hatk and vecc = hati - 2hatj + hatk` find a unit vector parallel to the vector `2veca - vecb + 3vecc`.

Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `P(2veca + vecb)` and `Q(veca - 3vecb)` externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

The two adjacent sides of a parallelogram are `2hati - 4hatj + 5hatk` and `hati - 2hatj - 3hatk`. Find the unit vector parallel to its diagonal. Also, find its area.

Let `veca = hati + 4hatj + 2hatk, vecb = 3hati - 2hatj + 7hatk ` and `vecc = 2hati - hatj + 4hatk`. Find a vector `vecd` which is perpendicular to both `veca` and `vecb`, and `vecc.vecd = 15`.

The scalar product of the vector `hati + hatj + hatk` with a unit vector along the sum of vectors `2hati + 4hatj - 5hatk` and  `lambdahati + 2hatj +  3hatk` is equal to one. Find the value of `lambda`.

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `vecc* vecd = 15` is equally inclined to `veca, vecb "and"  vecc.` 

Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`

If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.

Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.

The value of is `hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj)` is ______.

If θ is the angle between any two vectors `veca` and `vecb,` then `|veca.vecb| = |veca xx vecb|` when θ is equal to ______.