Mathematics - Vector Question with Solution | TestHub

Let vector $\mathbf{r}$ be coplanar to $\mathbf{a}$ and $\mathbf{b}$. $$\begin{array}{ll}\therefore & \mathbf{r}=\mathbf{a}+t\mathbf{b}\\ \Rightarrow & \mathbf{r}=(\hat{\mathbf{i}}+2\hat{\mathbf{j}}+\hat{\mathbf{k}})+t(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\hat{\mathbf{i}}(1+t)+\hat{\mathbf{j}}(2-t)+\hat{\mathbf{k}}(1+t)\end{array}$$ The projection of $\mathbf{r}$ on $\mathbf{c}=\frac{1}{\sqrt{3}}\Rightarrow \frac{\mathbf{r}\cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}}$ $\Rightarrow \frac{|1\cdot (1+t)+1\cdot (2-t)-1\cdot (1+t)|}{\sqrt{3}}=\frac{1}{\sqrt{3}}$ $\Rightarrow |2-t|=\pm 1\Rightarrow t=1$ or 3 When, $t=1$ we have $\mathbf{r}=2\hat{\mathbf{i}}+\hat{\mathbf{j}}+2\hat{\mathbf{k}}$ When, $t=3$ we have $\mathbf{r}=4\hat{\mathbf{i}}-\hat{\mathbf{j}}+4\hat{\mathbf{k}}$