Question

4)
CABC is a
parallelogram
BCD is a straight line.
BD-3BC
Mis the midpoint of Oc.
OR
590
AB-y
(a) Find, in terms of x and y.
JAM

Answer 1

Answer:

a(i) $\overrightarrow{AM}=\frac{1}{2}y-x$

 (ii) $\overrightarrow{OD}=y-2x$

(b)

• $\overrightarrow{OD}\ is\ parallel\ to\ \overrightarrow{AM}$
• $The\ magnitude\ of\ \overrightarrow{OD}\ is\ twice\ the\ magnitude\ of\ \overrightarrow{AM}$

Step-by-step explanation:

Finding the Vector of 2 Points with Given Vectors:

We can add the given vectors to find the vector of 2 points, providing the added vectors form a continuous line from the same Start Point & End Point with the asked vector.

(i)

$\overrightarrow{AM}=\overrightarrow{AN}+\overrightarrow{NM}$

• $\overrightarrow{AN}\ has\ \frac{1}{2}\ length\ and\ same\ direction\ with\ \overrightarrow{AB} \Longrightarrow\overrightarrow{AN}=\frac{1}{2} \overrightarrow{AB}=\frac{1}{2} y$
• $\overrightarrow{NM}\ has\ same\ length\ but\ opposite\ direction\ with\ \overrightarrow{OA}\Longrightarrow\overrightarrow{NM}=-\overrightarrow{OA}=-x$

$\overrightarrow{AM}=\frac{1}{2}y-x$

(ii)

$\overrightarrow{OD}=\overrightarrow{OC}+\overrightarrow{CD}$

• $\overrightarrow{OC}\ has\ same\ length\ and\ same\ direction\ with\ \overrightarrow{AB} \Longrightarrow\overrightarrow{OC}= \overrightarrow{AB}= y$
• $\overrightarrow{CD}\ has\ 2\times length\ but\ opposite\ direction\ with\ \overrightarrow{OA}\Longrightarrow\overrightarrow{CD}=-2\times\overrightarrow{OA}=-2x$

$\overrightarrow{OD}=y-2x$

(b)

$\overrightarrow{OD}=y-2x$

      $=2(\frac{1}{2} y-x)$

      $=2\times\overrightarrow{AM}$

Therefore:

• $\overrightarrow{OD}\ is\ parallel\ to\ \overrightarrow{AM}$
• $The\ magnitude\ of\ \overrightarrow{OD}\ is\ twice\ the\ magnitude\ of\ \overrightarrow{AM}$