The number of values of $m \in R$ for which the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+m…
The number of values of $m \in R$ for which the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+m \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+m \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are collinear is
2
3
1
infinite
Solution
Vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+m \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+m \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are collinear.
$
\begin{aligned}
\because \quad \mathbf{a} & =a_1 \hat{\mathbf{i}}+b_1 \hat{\mathbf{j}}+c_1 \hat{\mathbf{k}} \\
\mathbf{b} & =a_2 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}+c_2 \hat{\mathbf{k}} \text { are collinear }
\end{aligned}
$
Then, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Now, $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+m \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+m \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are collinear
$
\Rightarrow \quad \frac{1}{1}=\frac{2}{m}=\frac{m}{2}
$
After solving, we get
$
m=2
$
$\therefore$ Number of values of $m$ is 1