# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#9

Question

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