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# 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 ?

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

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• 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)$