Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#6

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Question

Two vectors  and  are such that   and  , where  is a constant.

     i.       Find the value of  for which  is perpendicular to .

   ii.       For the case where , find the angle between the directions of  and  .

Solution

     i.
 

If  and  & , then  and  are perpendicular.

Therefore, we need the scalar/dot product of  and  to equate it with ZERO.

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

Therefore;

Now we have two options;

   ii.
 

For the case where ;

We recognize that to find the angle between  and , we need the scalar/dot product of these two. We have;

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 ;

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