# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#4

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Question

Relative to the origin O, the position vectors of points A and B are given by and i.
Find the cosine of angle AOB.

The position vector of C is given by .

ii.       Given that AB and OC have the same length, find the possible values of k.

Solution

i.

We recognize that angle AOB is between and . Therefore, we need the scalar/dot product of  these two to find angle AOB.

Now we need to find .

The scalar or dot product of two vectors and in component form is given as;     Since ; We are given;  Therefore;   The scalar or dot product of two vectors and is number or scalar , where is  the angle between the directions of and . where  Therefore, for the given case; Therefore;  Hence;   Now we can equate the two equations of ;    ii.

We are given that; First we need to find .

A vector in the direction of is; For the given case; We are given;  Therefore;  We are given that and are of same length, that means; Magnitude of a vector ;  Therefore;          Evidently, it is a quadratic equation for which solution is; For the above equation;   Therefore;  Now we have two options.      