Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the required ratio be $\lambda :1$.

The position vector of the point dividing the line segment joining $\vec{a}=-2\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{b}=7\hat{i}-\hat{k}$ is given by:

$\vec{r}=\frac{\lambda \vec{b}+\vec{a}}{\lambda +1}$

Given $\vec{r}=\hat{i}+2\hat{j}+3\hat{k}$, we have:

$\hat{i}+2\hat{j}+3\hat{k}=\frac{\lambda (7\hat{i}-\hat{k})+(-2\hat{i}+3\hat{j}+5\hat{k})}{\lambda +1}$

Equating the coefficients of $\hat{i}$ on both sides:

$1=\frac{7\lambda -2}{\lambda +1}$

$\lambda +1=7\lambda -2$

$6\lambda =3$

$\lambda =\frac{1}{2}$

The ratio is $1:2$.

Answer: $1:2$