# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#6

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Question

Relative to an origin , the position vectors of points A and B are given by

 and

i.
Use a scalar product to find angle OAB.

ii.
Find the area of triangle OAB.

Solution

i.

It is evident that angle OAB is between  and .

We are given that;

 and

Next, we need scalar/dot product of  and .

However, first we need to find .

A vector in the direction of  is;

For the given case;

Now we can find 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;

Scalar/Dot product is also defined as below.

The scalar or dot product of two vectors  and  is number or scalar , where  is the  angle between the directions of  and  .

For;

Therefore, we need to find  and .

Expression for the length (magnitude) of a vector is;

Therefore;

Hence;

Equating both scalar/dot products found above;

Therefore;

ii.

We know length of two sides of desired triangle OAB as found in (i).

We have also found in (i) one of the angles of triangle OAB which included between sides  and  ;

Expression for the area of a triangle for which two sides (a and b) and the included angle (C ) is given;

Therefore;