Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#8

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Question

The position vectors of points A and B, relative to an origin O, are given by

and

Where  is a constant.

i.Find the value of  for which the angle AOB=90o.

ii.Find the values of  for which the lengths of OA and OB are equal.

The point C is such that .

iii.In the case where , find the unit vector in the direction of .

Solution


i.

It is evident that angle AOB is between and .

We are given that angle AOB is .

If and & , then and are perpendicular.

Therefore, we need scalar/dot product of and ;

We are given that;

and

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

Since ;

For the given case;


ii.

We are required to find the value of for which;

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

Therefore;

We are given that;

 

iii.

We are required to find unit vector in the direction of .

We are given that  and;

and

Therefore;

and

We are also given that;

A vector in the direction of  is;

Therefore;

We are required to find unit vector in the direction of .

A unit vector in the direction of  is;

Therefore, for the given case;

Therefore, we need vector and its magnitude .

We already have vector and need to find its magnitude .

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

Therefore;

Now we can find unit vector in the direction of .

 

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