Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#7

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Question

Relative to an origin O, the position vectors of the points A, B and C are given by

and


i.         
In the case where ABC is a straight line, find the values of p and q.

ii.       In the case where angle BAC is 90o, express q in terms of p.

 iii.   In the case where p=3 and the lengths of AB and AC are equal, find the possible values of q.

Solution

     i.
 

If three points A,B and C lie on the same straight line then two vectors starting from the the same  point, say A, but ending at both other different points, B or C,  must be parallel. .

Conversely if  then all three points A, B and C lie sin the same straight line.

We are given;

and

We are also given that ABC is a straight line, therefore, .

Lets first find  and .

A vector in the direction of  is;

For the given case;

Therefore;

Next we need to find .

A vector in the direction of  is;

For the given case;

Therefore;

Two vectors A and B are parallel if and only if they are scalar multiples of one another. If ,  then;

where k is a constant not equal to zero. 

Therefore, for the given case;

It is evident that if two vectors are scalar multiples then;

Therefore;

Substituting ;

We also have;

Substituting ;


ii.
 

We are given that angle BAC is 90o.

If  and  & , then  and  are perpendicular.

Therefore,  and  are perpendicular.
Hence  
.

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

Since ;

We are given that;

Hence;

  iii.
 

We have already found  and  in (i)

We are given that , therefore;

We are also given that the lengths of AB and AC are equal. Therefore;

Lets now find the lengths (magnitudes) of both  and  and equate them.

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

Hence;

Since ;

Now we have two options;

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