## Chapter 10 - Vector Algebra

#### Page 428

Represent graphically a displacement of 40 km, 30° east of north.

Classify the following measures as scalars and vectors.

(i) 10 kg

(ii) 2 metres north-west

(iii) 40°

(iv) 40 watt

(v) 10^{–19} coulomb

(vi) 20 m/s^{2}

Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done

In Figure, identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Answer the following as true or false.

Two vectors having same magnitude are collinear.

Answer the following as true or false.

`veca and -veca` are collinear

Answer the following as true or false.

Two collinear vectors are always equal in magnitude.

Answer the following as true or false.

Two collinear vectors having the same magnitude are equal

#### Pages 440 - 441

Compute the magnitude of the following vectors:

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

Write two different vectors having same magnitude.

Write two different vectors having same direction.

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 `bar(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`

Find a vector in the direction of vector `5hati - hatj +2hatk` which has magnitude 8 units.

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.

Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.

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, in the ration 2:1

(i) internally

(ii) externally

Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

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:

(A) AB + BC + CA = 0

(B) AB+ BC −AC = 0

(C) AB+ BC −CA = 0

(D) AB−CB+CA = 0

If a and b are two collinear vectors, then which of the following are incorrect:

(A) b = λa, for some scalar λ

(B) a = ±b

(C) the respective components of a and b are not proportional

(D) both the vectors a and b have same direction, but different magnitudes.

#### Pages 447 - 448

Find the angle between two vectors `veca` and `vecb` with magnitudes `sqrt3` and 2, respectively having `veca.vecb = sqrt6`.

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 `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)`

Also, show that they are mutually perpendicular to each other.

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*.

Show that `|veca|vecb+|vecb|` is perpendicular to `|veca|vecb-|vecb|veca`, for any two nonzero vectors `veca and vecb`

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

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

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 points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Show that the vectors `2hayti - hatj + hatk` and `3hati - 4hatj - 4hatk` `form 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

(A) λ = 1 (B) λ = –1

(C) a = |λ|

(D) a = 1/|λ|

#### Pages 454 - 455

Find |a ×b|, 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).

Find the area of the parallelogram whose adjacent sides are determined by the vector `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`

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

(A) `pi/6`

(B) `pi/4`

(C) `pi/3`

(D) `pi/2`

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` and respectively is

(A) 1/2

(B) 1

(C)2

(D) 4

#### Pages 458 - 459

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of *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)`

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.

If `veca = vecb + vecc`, then is it true that `|veca| = |vecb| + |vecc|`? Justify your answer.

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 `(2veca + vecb)` and `(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.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are .`1/sqrt3 , 1/sqrt3. 1/sqrt3`

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 `veca + vecb+ vecc` 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

(A) `0 < theta < pi/2`

(B) `0 <= theta <= pi/2`

(C) `0 < theta < pi`

(D) `0 <= theta <= pi`

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

(A) `theta = pi/4`

(B) `theta = pi/3`

(C) `theta =pi/2`

(D) `theta = 2pi/3`

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

(A) 0

(B) –1

(C) 1

(D) 3

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

(A) 0

(B) `pi/4`

(C) `pi/4`

(D) π