# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#7

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Question

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

 and

i.       Find .

ii.       The point M is the mid-point of AC. Find the unit vector in the direction of  .

iii.       Evaluate and hence find angle BAC.

Solution

i.

We are required to find the vector .

A vector in the direction of  is;

For the given case;

ii.

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

A unit vector in the direction of  is;

For the given case;

It is evident that first we need to find  and then

We are also given that point M is the mid-point of the AC. Therefore;

Next we need .

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

Therefore;

Hence;

iii.

We are required to find and then find angle BAC.

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

Since ;

Therefore, to find we need vectors and .

We have already found in (i).

Therefore, we need to find  first.

A vector in the direction of  is;

For the given case;

Now we can find  .

It is evident that angle BAC is between and .

We have found above that;

Scalar/Dot product is also defined as below.

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

For;

Therefore, we need to find  and .

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

Therefore;

Hence;

Equating both scalar/dot products found above;

Therefore;