# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#3

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Question

Relative to an origin , the position vectors of points A and B are given by  and  where is a constant.

i.       Find the value of for which angle AOB = .

ii.       In the case where , find the vector which has magnitude 28 and is in the same direction as .

Solution

i.

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

We are given that;  The scalar or dot product of two vectors and in component form is given as;       Since ;   Angle AOB= between and means and are perpendicular to each other.

If and & , then and are perpendicular.

Therefore, we need to equate scalar/dot product with ZERO.      ii.

We are given that;  When ;  A vector in the direction of is; Therefore to find ;    Expression for the length (magnitude) of a vector is;  For the given case; But we are looking for a vector in the direction of but with magnitude 28. Therefore desired vector in the direction of will be; A unit vector in the direction of is;   For the given case; Hence desired vector in the direction of ;   