Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2019  OctNov  (P19709/11)  Q#10
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Question
Relative to an origin O, the position vectors of points A, B, C and D, shown in the diagram, are given by are given by


and 


i. Show that AB is perpendicular to BC.
ii. Show that ABCD is a trapezium.
iii. Find the area of ABCD, giving your answer correct to 2 decimal places.
Solution
i.
We are required to show that AB is perpendicular to BC i.e., .
If and & , then and are perpendicular.
It is evident that first we need to show that scalar/dot product of vectors and equals ZERO.
First, we find .
A vector in the direction of is;
For the given case;
Next, we find .
Now we find the scalar/dot product of vector and .
The scalar or dot product of two vectors and in component form is given as;


Since ;
For the given case;
Since scalar/Dot product is vector and equals ZERO, the two vectors are perpendicular.
ii.
We are required to show that ABCD is a trapezium.
It is evident that ABCD will be a trapezium if AB is parallel to DC as we already have shown that AB is perpendicular to BC.
Therefore, we need to show that is parallel to .
The vectors and are parallel if, and only if, they are scalar multiples of one another:⃑
where is a nonzero real number.
Hence, we need to show that;
We have found in (ii) that;
Next, we find .
A vector in the direction of is;
For the given case;
Hence, and are parallel and, therefore, ABCD is a trapezium.
iii.
We are required to find the area of ABCD.
We have demonstrated in (ii) that ABCD is a trapezium.
Expression for area of trapezium for which height is and length of parallel sides are and ;
For the given case, since and are perpendicular, as shown in (i);
Hence;
We have found in (i) and (ii) that;
We need magnitudes of all , and .
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
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