Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#9

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Question

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

and

where  and  are constants. Find 
i.        
the unit vector in the direction of ,

ii.      the value of p for which angle ,

iii.       the values of  for which the length of   is 7 units.

Solution


i.
 

A vector in the direction of  is;

Therefore, for the given case;

A unit vector in the direction of  is;

For the given case;


ii.
 

We are given that angle  , hence,  and  are perpendicular and therefore their dot/scalar product is equal to ZERO.

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

Since ;


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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