# Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.6 | Set 1

• Last Updated : 28 Mar, 2021

### Question 1: Find the magnitude of the vector .

Solution:

Magnitude of a vector

=>

=>

=>

=>

### Question 2: Find the unit vector in the direction of .

Solution:

We know that unit vector of a vector  is given by,

=>

=>

=>

=>

=>

### Question 3: Find a unit vector in the direction of the resultant of the vectors ,  and .

Solution:

Let,

=>

=>

=>

Let  be the resultant,

=>

=>

=>

Unit vector is,

=>

=>

=>

=>

### Question 4: The adjacent sides of a parallelogram are represented by the vectors  and . Find the unit vectors parallel to the diagonals of the parallelogram.

Solution:

Let PQRS be the parallelogram.

Given that, PQ =  and QR = .

Thus, the diagonals are: PR and SQ.

=>

=>

=>

=>

=>

=>

=>

=>

Thus the unit vectors in the direction of the diagonals are:

=>

=>

=>

=>

=>

=>

### Question 5: If,  and , find .

Solution:

Given,  and .

Let,

=>

=>

=>

=>

The magnitude is given by,

=>

=>

=>

### Question 6: If  and the coordinates of P are (1,-1,2), find the coordinates of Q.

Solution:

Given,

And,

=>

=>

=>

=>

=> Thus the coordinates of Q are (4,1,1).

### Question 7: Prove that the points ,  and  are the vertices of a right-angled triangle.

Solution:

Let,

=>

=>

=>

Thus, the 3 sides of the triangle are,

=>

=>

=>

=>

=>

=>

=>

=>

=>

The lengths of every side are given by their magnitude,

=>

=>

=>

As we can see,

=>

=> These 3 points form a right-angled triangle.

### Question 8: If the vertices A, B and C of a triangle ABC are the points with position vectors , ,  respectively, what are the vectors determined by its sides? Find the length of these vectors.

Solution:

Let,

=>

=>

=>

The sides of the triangle are given as,

=>

=>

=>

=>

=>

=>

=>

=>

=>

The lengths of the sides are,

=>

=>

=>

### Question 9: Find the vector from the origin O to the centroid of the triangle whose vertices are (1,-1,2), (2,1,3), and (-1,2,-1).

Solution:

The position of the centroid is given by,

=> (x, y, z) =

=> (x, y, z) =

=> (x, y, z) =

The vector to the centroid from O is,

=>

### (i) Internally

Solution:

The position vectors of a point that divides a line segment internally are given by,

=> , where

=>

=>

### (ii) Externally

Solution:

The position vectors of a point that divides a line segment externally are given by,

=> , where

=>

=>

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