### Exercise 10.1

### Solve The Following Questions.

**Question****1. Represent graphically a displacement of 40 km, 30° east of north.**

**Solution :**

Displacement 40 km, **30°** East of North

Here, vector represents the displacement of 40 km, 30° East of North.

**Question****2. Check the following measures as scalars and vectors:**

**(i) 10 kg **

**(ii) 2 meters north-west **

**(iii) 40°**

**(iv) 40 Watt **

**(v) 10 ^{–19} coulomb **

**(vi) 20 m/sec ^{2}**

**Solution :**

(i) 10 kg is a measure of mass, it has no direction, it is magnitude only and therefore it is a scalar.

(ii) 2 meters North-West us a measure of velocity. It has magnitude and direction both and hence it is a vector.

(iii) 40° is a measure of angle. It has no direction, it has magnitude only. Therefore it is a scalar.

(iv) 40 Watt is a measure of power. It has no direction, only magnitude and therefore, it is a scalar.

(v) 10^{–19} coulomb is a measure of electric charge and it has magnitude only, therefore, it is a scalar.

(vi) 20 m/sec^{2} is a measure of acceleration. It is a measure of rate of change of velocity, therefore, it is a vector.

**Question****3. Classify the following as scalar and vector quantities:**

**(i) time period **

**(ii) distance **

**(iii) force**

**(iv) velocity **

**(v) work done**

**Solution :**

(i) Time-scalar

(ii) Distance-scalar

(iii) Force-vector

(iv) Velocity-vector

(v) Work done-scalar

**Question****4. In the adjoining figure, (a square) identify the following vectors:**

**(i) Coinitial**

**(ii) Equal**

**(iii) Collinear but not equal**

**Solution :**

(i) and have same initial point and therefore coinitial vectors.

(ii) and have same direction and same magnitude. Therefore and are equal vectors.

(iii) and have parallel support, so that they are collinear. Since they have opposite directions, they are not equal. Hence and are collinear but not equal.

**Question****5. Answer the following as true or false:**

**(i) and – are collinear.**

**(ii) Two collinear vectors are always equal in magnitude.**

**(iii) Two vectors having same magnitude are collinear.**

**(iv) Two collinear vectors having the same magnitude are equal.**

**Solution :**

(i) True. Vectors and-are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iv) False.

Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points.

Exercise 10.2

Solve The Following Questions.

**Question****1. Compute the magnitude of the following vectors:**

**Solution :**

The given vectors are:

**Question****2. Write two different vectors having same magnitude.**

**Solution :**

Hence, and are two different vectors having the same magnitude. The vectors are different because they have different directions.

**Question****3. Write two different vectors having same direction.**

**Solution :**

The direction cosines of and are the same. Hence, the two vectors have the same direction.

**Question4**.** Find the values of x and y so that the vectors are equal**

**Solution :**

The two vectors will be equal if their corresponding components are equal.

Hence, the required values of *x* and* y* are 2 and 3 respectively.

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

**Solution :**

The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

Hence, the required scalar components are –7 and 6 while the vector components are

**Question****6. Find the sum of the vectors: **

**Solution :**

Given:

**Question****7. Find the unit vector in the direction of the vector **

**Solution :**

The unit vector in the direction of vector is given by = / |a|

**Question****8. Find the unit vector in the direction of the vector where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.**

**Solution :**

Given: Points P (1, 2, 3) and Q (4, 5, 6)

Hence, the unit vector in the direction of is

**Question****9. For given vectors find the unit vector in the direction of + **

**Solution :**

Given: Vectors

Hence, the unit vector in the direction of (**+ ) **is

**Question****10. Find the vector in the direction of vector which has magnitude 8 units.**

**Solution :**

Let ** = **

Hence, the vector in the direction of vector which has magnitude 8 units is given by,

**Question****11. Show that the vectors are collinear.**

**Solution :**

Let

= λ

where λ = 2

Hence, the given vectors are collinear.

**Question****12. Find the direction cosines of the vector **

**Solution :**

The given vector is =

We know that the direction cosines of a vector are coefficients of

**Question****13. Find the direction cosines of the vector joining the points A (1, 2, –3) and** **B (–1, –2, 1) directed from A to B.**

**Solution :**

The given points are A (1, 2, –3) and B (–1, –2, 1).

**Question****14. Show that the vector is equally inclined to the axes OX, OY and OZ.**

**Solution :**

Let =

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

**Question****15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are and – respectively, in the ratio 2 : 1**

**(i) internally**

**(ii) externally.**

**Solution :**

The position vector of point R dividing the line segment joining two points

P and Q in the ratio *m:* *n *is given by

(i) Internally:

=

(ii) Externally:

=

Position vectors of P and Q are given as:

**Question****16. Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q** **(4, 1, – 2).**

**Solution :**

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

**Question****17. Show that the points A, B and C with position vectors respectively form the vertices of a right angled triangle.**

**Solution :**

Position vectors of points A, B, and C are respectively given as:

Hence, ABC is a right-angled triangle.

**Question****18. In triangle ABC (Fig. below), which of the following is not true:**

**Solution :**

Hence, the equation given in alternative C is **incorrect**.

The correct answer is **C.**

**Question****19. If and are two collinear vectors, then which of the following are incorrect:**

**(A) = λ for some scalar λ**

**(B) = ±**

**(C) The respective components of and are proportional.**

**(D) Both the vectors and have same direction, but different magnitudes.**

**Solution :**

Option (D) is not true because two collinear vectors can have different directions and also different magnitudes.

The option (A) and option (C) are true by definition of collinear vectors.

Option (B) is a particular case of option (A).

**Exercise 10.3**

Solve The Following Questions.**Question**** 1. Find the angle between two vectors and with magnitude √3 and 2 respectively having . = √6**

**Solution :**

It is given that,

Hence, the angle between the given vectors and** **is π/4.**Question ****2. Find the angle between the vectors .**

**Solution :**

The given vectors are**Question**** 3. Find the projection of the vector on the vector **

**Solution :**

Let = and ** = **

Now, projection of vector** **on** **is given by,

Hence, the projection of vector ** **on** **is 0.**Question ****4. Find the projection of the vector on the vector **

**Solution :**

Let ** = ** and **= **

Now, projection of vectoron** **is given by,

**Question**** 5. Show that each of the given three vectors is a unit vector:**

**Also show that they are mutually perpendicular to each other.****Solution :**

Hence, the given three vectors are mutually perpendicular to each other.

**Question**** 6. Find **

**Solution :****Question**** 7. Evaluate the product **

**Solution :****Question**** 8. Find the magnitude of two vectors and having the same magnitude such that the angle between them is 60° and their scalar product is 1/2.**

**Solution :**

Let *θ* be the angle between the vectors** and**

**Question ****9. Find if for a unit vector .**

**Solution :**

**Question**** 10. If ****are such that + λ is perpendicular to then find the value of λ**

**Solution :**

Given: **Question**** 11. Show that is perpendicular to for any two non-zero vectors and **

**Solution :****Question**** 12. If and . = 0 and . = 0 , then what can be concluded about the vector ?**

**Solution :**

It is given that. = 0 and . = 0 .

Hence, vectorsatisfying. = 0can be any vector.**Question**** 13. If , and are unit vectors such that + += 0 find the value of **

**Solution :**

Since, **+ += 0** are unit vectors.**Question**** 14. If either vector . But the converse need not be true. Justify your answer with an example.****Solution :**

Hence, the converse of the given statement need not be true.**Question ****15.** **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 ]****Solution :**

The vertices of ΔABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2).

Also, it is given that ∠ABC is the angle between the vectors.

**Question**** 16. ****Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.**

**Solution :**

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

Hence, the given points A, B, and C are collinear.**Question**** 17. Show that the vectors form the vertices of a right angled triangle.**

**Solution :**

Let vectors** ** be position vectors of points A, B, and C respectively.

Hence, ΔABC is a right-angled triangle.**Question**** 18. ****If is a nonzero vector of magnitude ‘ a’ and λ a nonzero scalar, then λ is unit vector if**

**(A) λ = 1**

**(B) λ = –1****(C) a = | λ |**

**(D) a = 1/|**

*λ|***Solution :**

Therefore, option (D) is correct.

**Exercise 10.4**

**Solve The Following Questions.**

**Question ****1. Find | x | if **

**Solution :**

We have,

**Question ****2. Find a unit vector perpendicular to each of the vectors **

**Solution :**

We have

**Question ****3. If a unit vector makes an angle π/3 with and an acute angle θ with then find θ and hence, the components of .**

**Solution :**

Let unit vector have (*a*_{1}, *a*_{2}, *a*_{3}) components.**Question**** 4. Show that **

**Solution :****5. Find λ and μ if ****Solution :****Question ****6. Given that . = 0 and x = 0 What can you conclude about the vectors and ?**

**Solution :**

. = 0

Then,

**(i)** Either || = 0 or || = 0, or ⊥(in case and are non – zero)

** x = 0**

**(ii)** Either || = 0 or || = 0 or ||(in case and are non – zero)

But, and cannot be perpendicular and parallel simultaneously.

Hence || = 0 or || =0.

**Question**** 7. Let the vectors ,, be given as **** then show that **

**Solution :**

We have**,**

Hence, the given result is proved.

**Question**** 8. It either = 0 and = 0 then x = 0 Is the converse true? Justify your answer with an example.**

**Solution :**

Take any parallel non-zero vectors so that** x **** = 0**

Hence, the converse of the given statement need not be true.

**Question**** 9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).**

**Solution :**

The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sides and of ΔABC are given as:

Hence, the area of ΔABC is √61/2 sq. units.**Question ****10. Find the area of the parallelogram whose adjacent sides are determined by the vectors **

**Solution :**

The area of the parallelogram whose adjacent sides are and is |x|.

Adjacent sides are given as:

Hence, the area of the given parallelogram is 15√2 sq. units.**Question ****11. Let the vectors and such that || = 3 and || = √2/3 then x is a unit vector, if the angle between and is:****(A) π/6****(B) π/4****(C) π/3****(D) π/2**

**Solution :**

It is given that |**| = 3 and |****| = √2/3**

We know that ** x = ||||sin ****θ **, where n is a unit vector perpendicular to both and and *θ* is the angle between ** **and.

Now, ** x **is a unit vector if **| x | = 1**

Therefore, option (B) is correct.**Question**** 12. Area of a rectangle having vertices A, B, C and D with position vectors respectively is:****(A) 1/2****(B) 1****(C) 2****(D) 4**

**Solution :**

The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

The adjacent sides and of the given rectangle are given as:

Now, it is known that the area of a parallelogram whose adjacent sides are ** and ** is **| x |**.

Hence, the area of the given rectangle is | x | = 2 sq. units.

Therefore, option (C) is correct.

**Miscellaneous Exercise**

### Solve The Following Questions.

**Question 1. Write down a unit vector in XY-plane making an angle of 30° with the positive direction of x-axis.**

**Solution :**

If is a unit vector in the XY-plane, then

Here, *θ* is the angle made by the unit vector with the positive direction of the *x*-axis.

Therefore, for *θ* = 30°:

**Question****2. Find the scalar components and magnitude of the vector joining the points **

**Solution :**

Given points are

Hence, the scalar components and the magnitude of the vector joining the given points are respectively

**Question 3. 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.**

**Solution :**

Let O and B be the initial and final positions of the girl respectively.

Then, the girl’s position can be shown as:

Hence, the girl’s displacement from her initial point of departure is

**Question 4. If then is it true that Justify your answer.**

**Solution :**

**Question 5. Find the value of x for which x () is a unit vector.**

**Solution :****x ()** is a unit vector if |**x ()| = 1**

Hence, the required value of *x* is ± 1/√3.

**Question 6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors **

**Solution :**

Given: Vectors

Let be the resultant of and

Hence, the vector of magnitude 5 units and parallel to the resultant of vectors and is

**Question 7. If find a unit vector parallel to the vector **

**Solution :**

Given: Vectors

**Question 8. 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.**

**Solution :**

The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).

Hence, point B divides AC in the ratio 2 : 3.

**Question 9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are (2 + ) and (- 3) externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.**

**Solution :**

It is given that .

It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:

Therefore, the position vector of point R is .

Position vector of the mid-point of RQ =

Hence, P is the mid-point of the line segment RQ.

**Question 10. Two adjacent sides of a parallelogram are Find the unit vector parallel to its diagonal. Also, find its area.**

**Solution :**

Adjacent sides of a parallelogram are given as:

Then, the diagonal of a parallelogram is given by ** + .**

Hence, the area of the parallelogram is 11√5 square units.

**Question 11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are **

**Solution :**

Let a vector be equally inclined to axes OX, OY, and OZ at angle *α*.

Then, the direction cosines of the vector are cos *α*, cos *α*, and cos *α*.

Hence, the direction cosines of the vector which are equally inclined to the axes are.

**Question 12. Let Find a vector which is perpendicular to both and and . = 15**

**Solution :**

**Question 13. The scalar product of the vector with a unit vector along the sum of vectors is equal to one. Find the value of λ.**

**Solution :**

Hence, the value of *λ* is 1.

**Question 14. If are mutually perpendicular vectors of equal magnitudes, show that the vector**** + + is equally inclined to ,, and.**

**Solution :**

Since,, andare mutually perpendicular vectors, we have

. = . = . = 0

It is given that:

| | = | | = | |

Let vector ** + + **be inclined to **,, **andat angles θ_{1} ,θ_{2} and θ_{3} respectively.

Then, we have:

Hence, the vector (** + + ) **is equally inclined to**,, **and.

**Question 15. Prove that if and only if , are perpendicular given .**

**Solution :**

**Question 16. Choose the correct answer:**

**If θ is the angle between two vectors and then .≥0 only when:**

**Solution :**

Let *θ* be the angle between two vectors and*.*

Then, without loss of generality, ** **and are non-zero vectors so that || and || are positives

It is known that .. = | |||cosθ

Therefore, option (B) is correct.

**Question 17. Choose the correct answer:**

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

**(A) θ = π/4**

**(B) θ = π/3**

**(C) θ = π/2 **

**(D) θ = π/3**

**Solution :**

Let and be two unit vectors and*θ* be the angle between them.

Then, || = || = 1

Now, + is a unit vector if | + | = 1

Hence, + is a unit vector if θ = 2π/3.

**Therefore, option (D) is correct.**

**Question 18. Choose the correct answer:**

**The value of is:**

**(A) 0 **

**(B) -1**

**(C) 1 **

**(D) 3**

**Solution :**

**Therefore, option (C) is correct.**

**Question 19. If θ be the angle between any two vectors and , then when ****θ is equal to:**

**(A) 0 **

**(B) π/4**

**(C) π/2 **

**(D) π**

**Solution :**

Let *θ* be the angle between two vectors ** and ***.*

Then, without loss of generality, ** and **are non-zero vectors, so that** **|| and || are positive.

Therefore, option (B) is correct.