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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
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Verified Answer
The correct answer is:
2
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.
$$
\begin{array}{lc}
\therefore & \frac{k-1}{2}=\frac{1}{2} \Rightarrow k-1=1 \\
\Rightarrow & k=2
\end{array}
$$
$$
\begin{aligned}
(k-1) \hat{\mathbf{i}}+(2 & -1) \hat{\mathbf{j}}+(3-2) \hat{\mathbf{k}} \\
= & (k-1) \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}...(i)
\end{aligned}
$$
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.
$$
\begin{array}{lc}
\therefore & \frac{k-1}{2}=\frac{1}{2} \Rightarrow k-1=1 \\
\Rightarrow & k=2
\end{array}
$$
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