Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/11)  Q#10
Question
Relative to an origin O, the position vectors of points A, B and C are given by


and 

respectively, where k is a constant.
i. Find the value of k in the case where angle AOB=90^{o}.
ii. Find the possible values of k for which the lengths of AB and OC are equal.
The point D is such that is in the same direction as and has magnitude 9 units. The point E is such that is in the same direction as and has magnitude 14 units.
iii. Find the magnitude of in the form where n is an integer.
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 given that;
We are given;
Expression for the length (magnitude) of a vector is;
Next we need to find .
A vector in the direction of is;
We are given that;

and 

Therefore, for the given case;
Expression for the length (magnitude) of a vector is;
Therefore, according to the given condition;
Now we have two options;






iii.
We are required to find the magnitude of .
Expression for the length (magnitude) of a vector is;
Therefore, first we need to find .
A vector in the direction of is;
Hence;
Now we first find .
We are given that and is in the same direction as .
A unit vector in the direction of is;
Conversely, a vector in a given direction is;
Therefore;
We are given that is in the same direction as . Therefore;
Let’s find unit vector of ;
A unit vector in the direction of is;
We are given;
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
Let’s now find .
We are given that and is in the same direction as .
A unit vector in the direction of is;
Conversely, a vector in a given direction is;
Therefore;
We are given that is in the same direction as . Therefore;
Let’s find unit vector of ;
A unit vector in the direction of is;
We are given;
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
Finally, we can write;
Now we can find magnitude of .
Expression for the length (magnitude) of a vector is;
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