# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#4

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Question

Relative to an origin O, the position vectors of points A, B and C are given by and The point P lies on AB and is such that .

i.
Find the position vector of P.

ii.
Find the distance OP.

iii.
Determine whether OP is perpendicular to AB. Justify your answer.

Solution

i.

We are given that; We need to find and .

A vector in the direction of is; Therefore;           ii.

Expression for the length (magnitude) of a vector is;  We have found in (i) that; Therefore; iii.

If and & , then and are perpendicular.

Therefore to determine whether OP is perpendicular to AB or not, we need to find out 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, we have found in (i) that; and    Since , vectors and are perpendicular.