Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.6 | Set 2

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,

=> 

=> 

=>

=> 

=> 

=>

=> 

=> 

=>

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