# Class 12 Maths NCERT Solutions for Chapter 10 Vector Algebra Miscellaneous Exercise

### Vector Algebra Miscellaneous Exercise Solutions

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

**Solution**

If r⃗ is a unit vector in the XY-plane, then r⃗ = cosÎ¸i ^ + sinÎ¸j ^ .

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

Therefore, for Î¸ = 30° .

r⃗ = cosÎ¸i ^ + sinÎ¸j ^ = (√3/2)i ^ + (1/2)j ^

Hence, the required unit vector is (√3/2) i ^ + (1/2)j ^.

**2. 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}) .**

**Solution**

The vector joining the points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) can be obtained by,

Hence, the scalar components and the magnitude of the vector joining the given points are respectively {(x_{2} - x_{1}), (y_{2} - y_{1}), (z_{2} - z_{1})}

and

**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 :

Now, we have:

By the triangle law of vector addition, we have :

Hence, the girl's displacement from her initial point of departure is (-5/2) i ^ + (3√3/2) j ^

**4. If a⃗ = b⃗ + c⃗, then is it true that |a⃗| = |b⃗| + |c⃗| ? Justify your answer.**

**Solution**

Now, by the triangle law of vector addition, we have a⃗ = b⃗ + c⃗.

It is clearly known that | a⃗|, |b⃗|, and |c⃗| represent the sides of Î”ABC.

∴ | a⃗| < |b⃗| + |c⃗|

Hence, it is not true that |a⃗| = |b⃗| + |c⃗| .

**5. Find the value of x for which x(i ^+ j ^+ k ^) is a unit vector.**

**Solution**

**6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a⃗ = 2i ^+3j ^-k^ and b⃗ = i ^- 2j ^+ k^ .**

**Solution**

Let c⃗ be the resultant of a⃗ and b⃗.

Then,

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

**7. If a⃗ = i ^+ j ^+ k^, b⃗ = 2i ^- j ^+3k^ and c⃗ = i ^- 2j ^+k^ find a unit vector parallel to the vector 2a⃗ - b⃗ +3c⃗ .**

**Solution**

**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**

Thus, the given points A, B, and C are collinear.

Now, let point B divide AC in the ratio Î» : 1. Then, we have :

On equating the corresponding components, we get :

5(Î» + 1) = 11Î» + 1

⇒ 5Î» + 5 =11Î» + 1

⇒ 6Î» = 4

⇒ Î» = 4/6 = 2/3

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

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

**Solution**

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.

**10 . The two adjacent sides of a parallelogram are 2i ^ - 4j ^ + 5k^, and i ^-2j ^-3k^ . Find the unit vector parallel to its diagonal. Also, find its area.**

**Solution**

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

∴ Area of parallelogram ABCD = | a⃗ + b⃗ |

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

**11. Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are 1/√3, 1/√3, 1/√3 .**

**Solution**

Then, the direction cosines of the vector are cos Î±, cos Î± and cos Î±.

cos

^{2}Î± + cos

^{2}Î± + cos

^{2}Î± = 1

⇒ 3cos

^{2}Î± = 1

⇒ cos Î± = 1/√3

Hence, the direction cosines of the vector which are equally inclined to the axes are 1/√3, 1/√3, 1/√3.

**12. Let a⃗ = i ^+4j ^+2k^ , b⃗=3i ^-2j ^+7k^ and c⃗ = 2i ^- j ^+4k^ . Find a vector**

**d⃗ which is perpendicular to both a⃗ and b⃗, and c⃗ . d⃗ = 15.**

**Solution**

Since d⃗ is perpendicular to both a⃗ and b⃗, we have :

**13. The scalar product of the vector i ^+ j ^+ k^, with a unit vector along the sum of vectors 2i ^+4j ^-5k^ and Î»i ^+2j ^+3k^ is equal to one. Find the value of Î» .**

**Solution**

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

**Solution**

Let vector a⃗+b⃗ +c⃗ be inclined to a⃗, b⃗ and c⃗ at angles Î¸

_{1}, Î¸

_{2}, Î¸

_{3}respectively .

Then , we have :

Now, as |a⃗|+|b⃗|+ |c⃗|, cos Î¸

_{1}= cos Î¸

_{2}= cosÎ¸

_{3}.

∴ Î¸

_{1}= Î¸

_{2}= Î¸

_{3}

Hence, the vector (a⃗ +b⃗ + c⃗) is equally inclined to a⃗, b⃗, and c⃗.

**15. Prove that (a⃗ +b⃗) . (a⃗ +b⃗) = |a⃗|**

^{2}+|b⃗|^{2}if and only if a⃗ . b⃗ are perpendicular, given a⃗ ≠ 0⃗ , b⃗ ≠ 0⃗**Solution**

**16. If Î¸ is the angle between two vectors a⃗ and b⃗, then a⃗ . b⃗ ≥ 0 only when**

(A) 0 < Î¸ < Ï€/2

(B) 0 ≤ Î¸ ≤ Ï€/2

(C) 0 < Î¸ < Ï€

(D) 0 ≤ Î¸ ≤ Ï€

(A) 0 < Î¸ < Ï€/2

(B) 0 ≤ Î¸ ≤ Ï€/2

(C) 0 < Î¸ < Ï€

(D) 0 ≤ Î¸ ≤ Ï€

**Solution**

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

Hence, a⃗ . b⃗ ≥ 0 when 0 ≤ Î¸ ≤ Ï€/2 .

The correct answer is B.

**17. Let a⃗ and b⃗ be two unit vectors and Î¸ is the angle between them. Then a⃗ + b⃗ is a unit vector if ____**

(A) Î¸ = Ï€/4

(B) Î¸ = Ï€/3

(C) Î¸ = Ï€/2

(D) Î¸ = 2Ï€/3

(A) Î¸ = Ï€/4

(B) Î¸ = Ï€/3

(C) Î¸ = Ï€/2

(D) Î¸ = 2Ï€/3

**Solution**

Now, a⃗ + b⃗ is a unit vector if |a⃗ +b⃗| = 1.

Hence, a⃗ + b⃗ is a unit vector if Î¸ = 2Ï€/3.

**18. The value of is**

**(A) 0**

(B) -1

(C) 1

(D) 3

(B) -1

(C) 1

(D) 3

**Solution**

The correct answer is C.

**19. If Î¸ is the angle between any two vectors a⃗ and b⃗, then |a⃗ . b⃗| = |a⃗ × b⃗| when Î¸ is equal to _____.**

(A) 0

(B) Ï€/4

(C) Ï€/2

(D) Ï€

(A) 0

(B) Ï€/4

(C) Ï€/2

(D) Ï€

**Solution**

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

Hence, | a⃗ . b⃗| = | a⃗ × b⃗| when Î¸ is equal to Ï€/4.