A vector vec{r} of magnitude 3sqrt{2} units w | vedclass

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(A) Let the vector be $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.The direction cosines are $l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$.Give

Vedclass · Accepted Answer

$\vec{r} = \pm 3\hat{i} + 3\hat{j}$

Vedclass · Answer

(A) Let the vector be $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.The direction cosines are $l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$.Given $\beta = \pi/4$ and $\gamma = \pi/2$, so $m = \cos(\pi/4) = 1/\sqrt{2}$ and $n = \cos(\pi/2) = 0$.Since $l^2 + m^2 + n^2 = 1$, we have $l^2 + (1/\sqrt{2})^2 + 0^2 = 1$, which gives $l^2 + 1/2 = 1$, so $l^2 = 1/2$, hence $l = \pm 1/\sqrt{2}$.The vector is $\vec{r} = |\vec{r}|(l\hat{i} + m\hat{j} + n\hat{k}) = 3\sqrt{2}(\pm \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + 0\hat{k}) = \pm 3\hat{i} + 3\hat{j}$.

$A$ vector $\vec{r}$ of magnitude $3\sqrt{2}$ units which makes angles of $\pi/4$ and $\pi/2$ respectively with $Y$ and $Z$ axes is:

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