If the line joining the points $(k, 2,3)$ and $(1,1,2)$ is parallel to the line joining the points $(5,4…
If the line joining the points $(k, 2,3)$ and $(1,1,2)$ is parallel to the line joining the points $(5,4,-1)$ and $(3,2,-3)$, then the value of $k$ is equal to
1
2
-2
3
Solution
Vector joining the points $(k, 2,3)$ and $(1,1,2)$ is
$
\begin{aligned}
(k-1) \hat{\mathbf{i}}+(2 & -1) \hat{\mathbf{j}}+(3-2) \hat{\mathbf{k}} \\
= & (k-1) \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}...(i)
\end{aligned}
$
and vector joining the two points $(5,4,-1)$
and
$
(3,2,-3) \text { is } 2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}...(ii)
$
Eqs. (i) and (ii) are parallel vector.
$
\begin{array}{lc}
\therefore & \frac{k-1}{2}=\frac{1}{2} \Rightarrow k-1=1 \\
\Rightarrow & k=2
\end{array}
$