Question

Find a vector of magnitude 5 units which is coplanar with vectors 3i^-j^-k^ and i^+j^-2k^ and is perpendicular to the vector 2i^+2j^-k^.

Answer 1

Dear student
$$\begin{gathered} Let\overrightarrow{a}=3i-j-k,\overrightarrow{b}=i+j-2kand\overrightarrow{c}=2i+2j+k \\ Anyvectorcoplanerwithvectors\overrightarrow{a}and\overrightarrow{b}is \\ \overrightarrow{r}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}where\lambda ,\mu aresomescalars. \\ \Rightarrow \overrightarrow{r}=\lambda \left(3i-j-k\right)+\mu \left(i+j-2k\right) \\ \Rightarrow \overrightarrow{r}=\left(3\lambda +\mu \right)i+\left(-\lambda +\mu \right)j+\left(-\lambda -2\mu \right)k...\left(1\right) \\ As\overrightarrow{r}isprependicularto\overrightarrow{c},\overrightarrow{r}.\overrightarrow{c}=0 \\ \Rightarrow \left(3\lambda +\mu \right).2+\left(-\lambda +\mu \right).2+\left(-\lambda -2\mu \right).1=0 \\ \Rightarrow 3\lambda +2\mu =0 \\ \Rightarrow \mu =-\frac{3}{2}\lambda \\ substitutingthisvalueof\mu in\left(1\right),weget \\ \overrightarrow{r}=\frac{3}{2}\lambda i-\frac{5}{2}\lambda j+2\lambda k....\left(2\right) \\ As\overrightarrow{r}isofmagnitude5units, \\ {\left(\frac{3}{2}\lambda \right)}^2+{\left(\frac{-5}{2}\lambda \right)}^2+{\left(2\lambda \right)}^2=5^2 \\ \Rightarrow \left(\frac{9}{4}+\frac{25}{4}+4\right){\lambda }^2=25 \\ \Rightarrow \frac{25}{2}{\lambda }^2=25 \\ \Rightarrow {\lambda }^2=2 \\ \Rightarrow \lambda =\pm \sqrt{2} \\ Substitutingthesevaluesineq\left(2\right)wegetrequiredvectors \\ =\pm \sqrt{2}\left(\frac{3}{2}i-\frac{5}{2}j+2k\right) \\ =\pm \frac{1}{\sqrt{2}}\left(3i-5j+4k\right) \end{gathered}$$
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