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

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

**Solution:**

Magnitude of

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

**Solution:**

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

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

**Solution:**

Let,

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Let be the resultant,

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Unit vector is,

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### 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.

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

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### Question 5: If, and , find .

**Solution:**

Given, , and .

Let,

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=>The magnitude is given by,

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

**Solution:**

Given,

And,

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=>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,

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Thus, the 3 sides of the triangle are,

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=>The lengths of every side are given by their magnitude,

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=>As we can see,

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=>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,

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The sides of the triangle are given as,

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=>The lengths of the sides are,

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### 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,

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### Question 10: Find the position vector of a point R which divides the line segment joining points p() and q() in the ratio 2:1.

### (i) Internally

**Solution:**

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

=> , where

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=>

### (ii) Externally

**Solution:**

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

=> , where

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=>