# 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;      