The quadrilateral $$$A B C D$$$ is a trapezium in which $$$A B$$$ and $$$D C$$$ are parallel. With respect to the origin $$$O$$$, the position vectors of $$$A, B$$$ and $$$C$$$ are given by $$$\overrightarrow{O A}=-\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}, \overrightarrow{O B}=\mathbf{i}+3 \mathbf{j}+\mathbf{k}$$$ and  $$$\overrightarrow{O C}=2 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}$$$ Find the distance between the parallel sides and hence find the area of the trapezium. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_33 Year：2021 Question No：9(c)

Answer:

For a general point $P$ on $A B$, state $\overrightarrow{C P}$ or $\overrightarrow{D P}$
in component form, e.g. $\overrightarrow{C P}=(3-2 \lambda,-\lambda,-6+2 \lambda)$
Equate a relevant scalar product to zero or equate derivative of $|\overrightarrow{C P}|$ to zero or use Pythagoras in a relevant triangle and solve for $\lambda$
Obtain $\lambda=2$
Show the perpendicular is of length 3
Carry out a correct method to find the area of $A B C D$ and obtain the answer 18
Alternative method for Question 9(c)
Use a scalar product to find the projection $C N$ (or $D N$ ) of $B C$ (or $A D$ ) on $C D$
Obtain $C N=3($ or $D N=3)$
Use Pythagoras to obtain $B N($ or $A N)$
Obtain answer 3
Carry out a correct method to find the area of $A B C D$ and obtain the answer 18

Knowledge points:

3.7.3 calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors (In 2 or 3 dimensions.)

3.7.6 use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points.

Answer:

Knowledge points:

Solution:

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