Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | May-Jun | (P1-9709/01) | Q#10

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Question

Relative to an origin , the position vectors of points  and  are    and   respectively.


i.       
Find the value of  for which  and  are perpendicular.

   ii.       In the case where , use a scalar product to find angle , correct to the nearest degree.

  iii.       Express the vector  in terms of  and hence find the values of  for which the length of  is 3.5 units.

Solution


i.
 

A point  has position vector from the origin . Then the position vector of  is denoted by  or .

If  and  & , then  and  are perpendicular.

Therefore, we need 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;


ii.
 

We are given that

Therefore;

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 ;

Therefore for the given case;

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

Where

For the given case;

Therefore;

Equating both scalar/dot products we get;


iii.
 

We are given that;

A vector in the direction of  is;

Therefore, for the given case;

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

Therefore;

Now we have two options;

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