# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | 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;