If $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}+3…
If $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{c}$ is a vector perpendicular to $\mathbf{b}$, then $\left\{\frac{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}{|\mathbf{b} \times \mathbf{c}|^2}\right\}(\mathbf{b} \times \mathbf{c})+\left\{\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}\right\} \mathbf{b}+\left\{\frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{c}|^2}\right\} \mid=$
$\sqrt{14}$
14
13
$\sqrt{17}$
Solution
Any vector $\mathbf{r}$ can be written in linear combination of two non-parallel vector $\mathbf{b}$ and $\mathbf{c}$, as
$
\mathbf{r}=\left(\frac{\mathbf{r} \cdot \mathbf{b}}{|\mathbf{b}|^2}\right) \mathbf{b}+\left(\frac{\mathbf{r} \cdot \mathbf{c}}{|\mathbf{c}|^2}\right) \mathbf{c}+\left(\frac{\mathbf{r} \cdot(\mathbf{b} \times \mathbf{c})}{|\mathbf{b} \times \mathbf{c}|^2}\right)(\mathbf{b} \times \mathbf{c})
$
So, $\mathbf{r}=\mathbf{a} \Rightarrow|\mathbf{r}|=|\mathbf{a}|=\sqrt{1+4+9}=\sqrt{14}$