Kinematics Question 148
Question: With respect to a rectangular cartesian coordinate system, three vectors are expressed as $ \vec{a}=4\hat{i}-\hat{j} $ , $ \vec{b}=-3\hat{i}+2\hat{j} $ and $ \vec{c}=-\hat{k} $ where $ \hat{i},\hat{j},\hat{k} $ are unit vectors, along the X, Y and Z-axis respectively. The unit vectors $ \hat{r} $ along the direction of sum of these vector is [Kerala CET (Engg.) 2003]
Options:
A) $ \hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k}) $
B) $ \hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}-\hat{k}) $
C) $ \hat{r}=\frac{1}{3}(\hat{i}-\hat{j}+\hat{k}) $
D) $ \hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}+\hat{k}) $
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Answer:
Correct Answer: A
Solution:
$ \vec{r}=\vec{a}+\vec{b}+\vec{c} $
$ =4\hat{i}-\hat{j}-3\hat{i}+2\hat{j}-\hat{k} $
$ =\hat{i}+\hat{j}-\hat{k} $
$ \hat{r}=\frac{{\vec{r}}}{|r|}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{1^{2}+1^{2}+{{(-1)}^{2}}}}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}} $
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