Question

Two vectors are given by $A=\hat{i}+2\hat{j}+2\hat{k}$ and $B=3\hat{i}+6\hat{j}+2\hat{k}$. Another vector $\overrightarrow{C}$ has the same magnitude as $\overrightarrow{B}$ but has the same direction as $\overrightarrow{A}$ . Then which of the following vectors represents $\overrightarrow{C}$?

• A. $\frac{7}{3}(\hat{i}+2\hat{j}+2\hat{k})$
• B. $\frac{7}{3}(\hat{i}-2\hat{j}+2\hat{k})$
• C. $\frac{7}{9}(\hat{i}-2\hat{j}+2\hat{k})$
• D. $\frac{9}{7}(\hat{i}+2\hat{j}+2\hat{k})$

Answer 1

Answer: (A)

$\frac{7}{3}(\hat{i}+2\hat{j}+2\hat{k})$

The vector $\overrightarrow{A}$ is given as $\overrightarrow{A}=\hat{i}+2\hat{j}+2\hat{k}$

Its magnitude $|A|=\sqrt{1^2+2^2+2^2}=3$

Unit vector along direction of $\overrightarrow{A}$, $\hat{A}=\frac{\overrightarrow{A}}{|A|}=\frac{\hat{i}+2\hat{j}+2\hat{k}}{3}$

Magnitude of $\overrightarrow{B}$, $|B|=\sqrt{3^2+6^2+2^2}=7$

Thus vector $\overrightarrow{C}$ is $\overrightarrow{C}=|B|\hat{A}$

$⟹$ $\overrightarrow{C}=\frac{7}{3}(\hat{i}+2\hat{j}+2\hat{k})$