Question

Find a vector in the direction of vector $\overrightarrow{a}=\stackrel{\wedge }{i}-2\stackrel{\wedge }{j}$ that has magnitude $7$ units.

• A. $\frac{7}{\sqrt{5}}(\stackrel{\wedge }{i}-2\stackrel{\wedge }{j})$
• B. $\frac{7}{\sqrt{5}}(\stackrel{\wedge }{i}+2\stackrel{\wedge }{j})$
  
• C. $\frac{7}{\sqrt{5}}(-\stackrel{\wedge }{i}-2\stackrel{\wedge }{j})$
  
• D. None of these

Answer 1

Answer: (A)

$\frac{7}{\sqrt{5}}(\stackrel{\wedge }{i}-2\stackrel{\wedge }{j})$

$\overrightarrow{a}=\hat{i}-2\hat{j}=1\hat{i}-2\hat{j}+0\hat{k}$

Magnitude of

$\begin{array}{c}\overrightarrow{a}=\sqrt{1^2+{\left(-2\right)}^2+0^2}\\ \left|\overrightarrow{a}\right|=\sqrt{1+4+0}\\ =\sqrt{5}\end{array}$

Unit vector in direction of

$\begin{array}{c}\hat{a}=\frac{1}{magnitudeof\overrightarrow{a}}\times \overrightarrow{a}\\ \hat{a}=\frac{1}{\sqrt{5}}\left[1\hat{i}+2\hat{j}+0\hat{k}\right]\\ \hat{a}=\frac{1}{\sqrt{5}}\hat{i}-\frac{2}{\sqrt{5}}\hat{j}+0\hat{k}\\ \hat{a}=\frac{1}{\sqrt{5}}\hat{i}-\frac{2}{\sqrt{5}}\hat{j}\end{array}$

Thus, Vector having a magnitude

$1=\frac{1}{\sqrt{5}}\hat{i}-\frac{2}{\sqrt{5}}\hat{j}$

Vector having a magnitude

$\begin{array}{c}7=7\left[\frac{1}{\sqrt{5}}\hat{i}-\frac{2}{\sqrt{5}}\hat{j}\right]\\ =\frac{7}{\sqrt{5}}\hat{i}-\frac{14}{\sqrt{5}}\hat{j}\end{array}$

$=\frac{7}{\sqrt{5}}\left(\hat{i}-2\hat{j}\right)$ (By taking common $\frac{7}{\sqrt{5}}$)