# Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.5

**Question 1. If the position vector of a point (-4,-3) be **** , find ****.**

**Solution:**

We have,

**Question 2. If the position vector **** of a point (12,n) is such that **** , find the value(s).**

**Solution:**

We have,

On squaring both sides,

**Question 3. Find a vector of magnitude 4 units which is parallel to the vector** .

**Solution: **

Given,

Let is a vector parallel to

Therefore,

for any scalar

**Question 4. Express in terms of unit vectors (i)A = (4,-1),B = (1,3) (ii)A = (-6,3) , B = (-2,-5)**

**Solution:**

(i) We have,

A = (4,-1)

B = (1,3)

Position Vector of A =

Position Vector of B =

Now,

Therefore,(ii) We have,

A = (-6,3)

B = (-2,-5)

Position Vector of A =

Position Vector of B =

Now,

Therefore,

**Question 5. Find the coordinates of the tip of the position vector which is equivalent to ****, where the coordinates of A and B are (-1,3) and (-2,1)**

**Solution:**

We have,

A = (-1,3)

B = (-2,1)

Now,

Position Vector of

Position Vector of

Therefore,

Coordinate of the position vector

**Question 6. ABCD is a parallelogram. If the coordinates of A,B,C are (-2,-1), (3,0),(1,-2) respectively, find the coordinates of D.**

**Solution:**

Here, A = (-2,-1)

B = (3,0)

C = (1,-2)

Let us assume D be (x , y).

Computing Position Vector of AB, we have,

= Position Vector of B – Position Vector of A

Comparing LHS and RHS of both,

5 = 1-x

x = -4

And,

1 = -2-y

y = -3

So, coordinates of D = (-4,-3).

**Question 7. If the position vectors of the points A(3,4), B(5,-6) and C(4,-1) are ****respectively, compute the value of ****.**

**Solution:**

Computing the position vectors of all the points we have,

Now,

Computing the final value after substituting the values,

**Question 8. If be the position vector whose tip is (-5,3), find the coordinates of a point B such that , the coordinates of A being (-4,1).**

**Solution:**

Given,

Coordinate of A = (4,-1)

Position vector of A =

Position vector of

Let coordinate of point B = (x, y)

Position vector of B =

Given that,

Position vector of B – Position vector of A = \vec{a}

Comparing the coefficients of LHS and RHS

x – y = 5

x = 9

Also,

y + 1 = 3

y = -1

So, coordinate of B = (9,-4)

**Question 9. Show that the points **** form an isosceles triangle. **

**Solution:**

So, the two sides AB and AC of the triangle ABC are equal.

Therefore, ABC is an isosceles triangle.

**Question 10. Find a unit vector parallel to the vector **** .**

**Solution:**

We have,

Let

Suppose is any vector parallel to

, where λ is any scalar.

Unit vector of

Therefore,

**Question 11. Find the components along the coordinate axes of the position vector of each of the following points : **

**(i) P(3,2)**

**(ii) Q(-5,1)**

**(iii) R(-11,-9)**

**(iv) S(4,-3)**

**Solution:**

(i) Given, P = (3,2)

Position vector of P =

Component of P along x-axis =

Component of P along y-axis =

(ii) Given, Q = (-5,1)

Position vector of Q =

Component of Q along x-axis =

Component of Q along y-axis =

(iii) Given, R = (-11,-9)

Position vector of R =

Component of R along x-axis =

Component of R along y-axis =

(iv) Given, S = (4,-3)

Position vector of S =

Component of S along x-axis =

Component of S along y-axis =