A line passes through the points with position vectors are $\hat i+\hat j-2\hat k$ and $\hat i-3\hat j+\hat k$. The position vector of a point on the line which is at a unit distance from the first point is ?

Question

$\begin{array}{1 1} (a)\:\frac{1}{5}(6\hat i+\hat j-7\hat k)\:\qquad\:(b)\:\frac{1}{5}(4\hat i+9\hat j-13\hat k)\:\qquad\:(c)\:\frac{1}{5}(\hat i-4\hat j+3\hat k)\:\qquad\:(d)\:None\:of\:these. \end{array} $

Answer 1

Toolbox:

• Eqn. of aline through two points $\overrightarrow a$ and $\overrightarrow b$ is $\overrightarrow r=\overrightarrow a+\lambda(\overrightarrow a-\overrightarrow b)$

Given points are $A(1,1,-2)$ and $B(1,-3,1)$

Equation of the line $AB$ is $\overrightarrow r=(\hat i+\hat j-2\hat k)+\lambda (4\hat j-3\hat k)$

Let the point be $C$ which is at a unit distance from $A$

Any point on the line $AB$ is given by $ \hat i+(1+4\lambda)\hat j-(2+3\lambda)\hat k$

Let this point be $C$.

Since $C$ is at unit distance from $A$, $ 16\lambda^2+9\lambda^2=1$

$\lambda=\large\frac{1}{5}$

$\therefore $ $C(\hat i+\large\frac{9}{5}$$\hat j-\large\frac{13}{5}$$\hat k)$

$i.e.,$ $ \large\frac{1}{5}$$(5\hat i+9\hat j-13\hat k)$