Practicing Success
A vector $\vec{r}$ has length 21 and direction ratios are proportional to 2, -3, 6. If $\vec{r}$ makes an acute angle with x-axis, then $\vec{r}$ = |
$6\hat{i}+9\hat{j} - 18 \hat{k}$ $6\hat{i}-9\hat{j} - 18 \hat{k}$ $6\hat{i}-9\hat{j} + 18 \hat{k}$ none of these |
$6\hat{i}-9\hat{j} + 18 \hat{k}$ |
Recall that if the direction ratios of a vector are proportional to a, b, c, then its direction cosines are $± \frac{a}{\sqrt{a^2+b^2+c^2}}, ± \frac{b}{\sqrt{a^2+b^2+c^2}}, ± \frac{c}{\sqrt{a^2+b^2+c^2}}$ Therefore, direction cosines of $\vec{r}$ are $± \frac{2}{\sqrt{2^2+(-3)^2+6^2}}, ± \frac{b}{\sqrt{2^2+(-3)^2+6^2}}, ± \frac{c}{\sqrt{2^2+(-3)^2+6^2}}$ Since $\vec{r}$ makes an acute angle with x-axis. $∴ cos \alpha > 0 \, i.e. l > o $ So, direction cosines of $\vec{r}$ are $\frac{2}{7}, -\frac{3}{7}, \frac{6}{7}$ $∴ \vec{r}= 21\left(\frac{2}{7} \hat{i} - \frac{3}{7} \hat{j} +\frac{6}{7}\hat{k}\right) $ $[∵ \vec{r} = |\vec{r}|(l \hat{i} + m \hat{j} + n \hat{k})]$ $⇒ \vec{r}=6\hat{i}-9\hat{j} + 18 \hat{k}$ |