Question

The diagram shows a scalene triangle, $ABC$, with vectors $\overrightarrow{AB}=3\hat{a}$ and $\overrightarrow{BC}=2\hat{b}$. The point $M$ is the midpoint of $AC$.Find an expression for the vector $\overrightarrow{AM}$ in terms of $\hat{a}$ and $\hat{b}$.

• A. $(\hat{a}+\frac{3}{2}\hat{b})$
• B. $(\hat{a}+\hat{b})$
• C. $(\frac{2}{3}\hat{a}+\hat{b})$
• D. $(\frac{3}{2}\hat{a}+\hat{b})$

Answer 1

The correct option is D $(\frac{3}{2}\hat{a}+\hat{b})$

First we can find the vector $\overrightarrow{AC}$
$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=3\hat{a}+2\hat{b}$
Since, $M$ is the midpoint of $AC$ we know that $\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AC}$

Thus, $\overrightarrow{AM}=\frac{1}{2}(3\hat{a}+2\hat{b})=(\frac{3}{2}\hat{a}+\hat{b})$