# RD Sharma Class 12 Ex 23.6 Solutions Chapter 23 Algebra of Vectors

Here we provide RD Sharma Class 12 Ex 23.6 Solutions Chapter 23 Algebra of Vectors for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 12 Ex 23.6 Solutions Chapter 23 Algebra of Vectors book pdf download. Now you will get step-by-step solutions to each question.

## RD Sharma Class 12 Ex 23.6 Solutions Chapter 23 Algebra of Vectors

### 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.

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=>

=>

=>

=>

=>

=>

=>

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,

=>

=>

=>

=>

=>

=>

=>

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=>

The lengths of every side are given by their magnitude,

=>

=>

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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

=>

=>

Question 11: Find the position vector of the mid-point of the vector joining the points P() and Q().
Solution:

The mid-point of the line segment joining 2 vectors is given by:
=>
=>
=>
=>
Question 12: Find the unit vector in the direction of the vector , where P and Q are the points (1,2,3) and (4,5,6).
Solution:
Let,
=>
=>
=>
=>
=>
Unit vector is,
=>
=>
=>
=>
Question 13: Show that the points A(), B(), C() are the vertices of a right-angled triangle.
Solution:
Let,
=>
=>
=>
The line segments are,
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=>
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=>
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The magnitudes of the sides are,
=>
=>
=>
As we can see that
=> Thus, ABC is a right-angled triangle.
Question 14: Find the position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).
Solution:
Let,
=>
=>
The mid-point of the line segment joining 2 vectors is given by:
=>
=>
=>
=>
Question 15: Find the value of x for which x() is a unit vector.
Solution:
The magnitude of the given vector is,
=>
=>
=>
For it to be a unit vector,
=>
=>
=>
Question 16: If  and , find a unit vector parallel to .
Solution:
Given,  and
=>
=>
Thus, the unit vector is,
=>
=>
=>
Question 17: If  and , find a vector of magnitude 6 units which is parallel to the vector .
Solution:
Given,  and
=>
=>
Unit vector in that direction is,
=>
=>
=>
Given that the vector has a magnitude of 6,
=> Required vectors are :  =
Question 18: Find a vector of magnitude 5 units parallel to the resultant of the vector and .
Solution:
Given,  and
The resultant vector will be given by,
=>
=>
=>
Unit vector is,
=>
=>
=>
Given that the vector has a magnitude of 5,
=> Required vectors are:
Question 19: The two vectors  and  represent the sides  and  respectively of the triangle ABC. Find the length of the median through A.
Solution:
Let D be the point on BC, on which the median through A touches.
D is also the mid-point of BC.
The median  is thus given by:
=>
=>
=>
=>
=>
=>
Thus, the length of the median is,
=>
=>
=>  units

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