Question

The position vectors of points A, B and C relative to an origin O are given by

where p is a constant.

i.       Find the value of p for which the lengths of AB and CB are equal.

ii.       For the case where p=1, use a scalar product to find angle ABC.

Solution

     i.
 

We are given that;

First we need to find .

A vector in the direction of  is;

We are given that;

Therefore, for the given case;

Expression for the length (magnitude) of a vector is;

Next we need to find .

A vector in the direction of  is;

We are given that;

Therefore, for the given case;

Expression for the length (magnitude) of a vector is;

Therefore, according to the given condition;

   ii.
 

It is evident that angle ABC is between  and . It is also quite fine to visualize the angle ABC  between  and .

From (i) we have found that;

Since p=1;

Next, we need scalar/dot product of  and .

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

For the given case;

Scalar/Dot product is also defined as below.

The scalar or dot product of two vectors  and  is number or scalar , where  is the  angle between the directions of  and  . 

For ;

We have found from (i) that;

Since p=1;

Equating both scalar/dot products found above;

Therefore;