Vectors 2 Question 17
17. If $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are any two non-collinear unit vectors and $\overrightarrow{\mathbf{a}}$ is any vector, then
$(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{b}}+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{c}}+\frac{\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|^{2}}(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=\ldots \ldots$
(1996, 2M)
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Answer:
Correct Answer: 17. $\sqrt{13}$ sq units 20. True
Solution:
- Let $\hat{\mathbf{i}}$ be a unit vector in the direction of $\overrightarrow{\mathbf{b}}, \hat{\mathbf{j}}$ in the direction of $\overrightarrow{\mathbf{c}}$. Note that $\overrightarrow{\mathbf{c}}=\hat{\mathbf{j}}$
and
$$ (\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=|\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}| \sin \alpha \hat{\mathbf{k}}=\sin \alpha \hat{\mathbf{k}} $$
where, $\hat{\mathbf{k}}$ is a unit vector perpendicular to $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$.
$$ \Rightarrow \quad|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|=\sin \alpha \Rightarrow \hat{\mathbf{k}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|} $$
Let
$$ \overrightarrow{\mathbf{a}}=a _1 \hat{\mathbf{i}}+a _2 \hat{\mathbf{j}}+a _3 \hat{\mathbf{k}} $$
Now,
$$ \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}=\hat{\mathbf{i}} \cdot\left(a _1 \hat{\mathbf{i}}+a _2 \hat{\mathbf{j}}+a _3 \hat{\mathbf{k}}\right)=a _1 $$
and
$$ \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{j}} \cdot\left(a _1 \hat{\mathbf{i}}+a _2 \hat{\mathbf{j}}+a _3 \hat{\mathbf{k}}\right)=a _2 $$
and $\quad \overrightarrow{\mathbf{a}} \cdot \frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|}=\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{k}}=a _3$
$$ \begin{aligned} & \therefore(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{b}}+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{c}}+\frac{\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|^{2}}(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}) \\ & \quad=a _1 \overrightarrow{\mathbf{b}}+a _2 \overrightarrow{\mathbf{c}}+a _3 \frac{(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|}=a _1 \hat{\mathbf{i}}+a _2 \hat{\mathbf{j}}+a _3 \hat{\mathbf{k}}=\overrightarrow{\mathbf{a}} \end{aligned} $$
