Question

The unit vector in ZOX plane, making angles \(45^{\circ}\) and \(60^{\circ}\) respectively with \(\vec{\alpha}=2 \hat{i}+2 \hat{j}-\hat{k}\) and \(\vec{\beta}=\hat{j}-\hat{k}\) is
(A) \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}\)
(B) \(\frac{1}{\sqrt{2}} \hat{i}-\frac{1}{\sqrt{2}} \hat{k}\)
(C) \(\frac{1}{\sqrt{2}} \hat{i}-\frac{1}{\sqrt{2}} \hat{j}\)
(D) \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}\)

Answer 1

Ans: (B)
Hint: Let the vector be \(\vec{r}=x \hat{i}+z \hat{k} \Rightarrow|\vec{r}|=1\)
\(\vec{r} \cdot \vec{\alpha}=|\vec{r}||\vec{\alpha}| \cos 45^{\circ}\)
\(\therefore 2 x-z=\frac{3}{\sqrt{2}}\)
\(\vec{r} \cdot \vec{\beta}=|\vec{r}| \vec{\beta} \mid \cos 60^{\circ}\)
\(z=-\frac{1}{\sqrt{2}}\)
\(\therefore x=\frac{1}{\sqrt{2}}\)
\(\therefore \overrightarrow{\mathrm{r}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}-\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\)

Answer 2

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