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

Hits: 875

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.

Comments