Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#5

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Question

Two vectors,  and , are such that

and

Where is a constant.

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

  ii.     Find the angle between  and when q = 0.

Solution


i.
 

We are given that;

If  and & , then  and  are perpendicular.

Therefore, if  and are perpendicular then

We next find the scalar/dot product of  and and equate that equal to zero.

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

Since ;

Since we are given  .

Standard form of quadratic equation is;

Its solution is given as;

Therefore, for the given case;

Now we have two options.

 

   ii.
 

Now we are required to find the angle between  and  if .

We are given that;

In the case where ,

To find the angle between  and  we need to find dot/scalar product of  and .

We are given that;

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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