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;    