# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#6

Hits: 692

Question

Relative to an origin , the position vectors of points and are given by      i.
Find the value of and hence state whether angle AOB is acute, obtuse or a right angle.

ii.       The point is such that . Find the unit vector in the direction of OX.

Solution

i.

We recognize that is angle between and .
Hence we use
scalar/dot product of and . We have both and .  The scalar or dot product of two vectors and in component form is given as;        Since ; For the given case;  If and & , then and are perpendicular.

If , then angle between and is acute.

If , then angle between and is obtuse.

Therefore angle AOB is obtuse.

ii.

From the given condition; We need to substitute in above equation to find .

A vector in the direction of is; Therefore, for the given case;   Hence, we can write;  A unit vector in the direction of is;   Now, unit vector in the direction of can be found as follows. Hence, we need . This can be found from as follows.    We can find as follows;   