1.7: Intersecting cevians using vectors

4PM1/1R/November/2020 — Question 11 · 13 marks

Figure 1 shows a triangle

\displaystyle OXY

\overrightarrow{OX}=2\mathbf{a} \qquad \text{and} \qquad \overrightarrow{OY}=3\mathbf{b}.

\displaystyle A

is the midpoint of

\displaystyle OX

and

\displaystyle B

is the point on

\displaystyle OY

such that

\displaystyle OB:BY=1:2

. The lines

\displaystyle XB

and

\displaystyle AY

intersect at

\displaystyle Z

(a) Find

\displaystyle \overrightarrow{AB}

as a simplified expression in terms of

\displaystyle \mathbf{a}

and

\displaystyle \mathbf{b}

(1)

(b) Using a vector method, find

\displaystyle \overrightarrow{OZ}

as a simplified expression in terms of

\displaystyle \mathbf{a}

and

\displaystyle \mathbf{b}

(9)

The point

\displaystyle M

\displaystyle XY

is such that

\displaystyle O

\displaystyle Z

and

\displaystyle M

are collinear.

\displaystyle \overrightarrow{OM}

as a simplified expression in terms of

\displaystyle \mathbf{a}

and

\displaystyle \mathbf{b}

(3)