Question

If $\mathbf{a}$ and $\mathbf{b}$ are position vectors of point A and point B, respectively, find the position vector of point C on $\overrightarrow{BA}$ such that $BC = 3BA$.

Hint:

When a point divides a vector in a given ratio, use the concept of weighted averages to find the position vector.

Answer 1

Let the position vectors of points A and B be $\mathbf{a}$ and $\mathbf{b}$, respectively. The vector $\overrightarrow{BA}$ is given by: \[ \overrightarrow{BA} = \mathbf{a} - \mathbf{b}. \] We are asked to find the position vector of point C such that $BC = 3BA$. The vector $\overrightarrow{BC}$ is given by: \[ \overrightarrow{BC} = \mathbf{c} - \mathbf{b}. \] Since $BC = 3BA$, we have: \[ \mathbf{c} - \mathbf{b} = 3(\mathbf{a} - \mathbf{b}). \] Simplifying: \[ \mathbf{c} - \mathbf{b} = 3\mathbf{a} - 3\mathbf{b}. \] Thus, the position vector of point C is: \[ \mathbf{c} = 3\mathbf{a} - 2\mathbf{b}. \]