# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#9

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Question

Relative to an origin , the position vectors of points A and B are given by  and  i.      Use a vector method to find angle AOB.

The point C is such that .

ii.       Find the unit vector in the direction of iii.     Show that triangle OAC is isosceles.

Solution

i.

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

We are given that;  The scalar or dot product of two vectors and in component form is given as;       Since ;    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.

A unit vector in the direction of is;   Therefore for the given case; Therefore, we need to find .

We are given that; A vector in the direction of is; We are given that;  Therefore;          Hence, unit vector in the direction of (or parallel to) ;   iii.

An isosceles triangle is a triangle with two equal sides.

We are to show that OAC is an isosceles triangle. If OAC is an isosceles triangle then either  A=OC or AC=OC.

Let us try to show OA=OC.

We are given that; We have found in (ii); We can find magnitudes of both vectors.

Expression for the length (magnitude) of a vector is;    Hence OAC is an isosceles triangle.