The projections of a vector on the three coor | vedclass

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(A) Let the vector be $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$.Given that the projections of the vector on the coordinate axes are $6, -3, 2$,we hav

Vedclass · Accepted Answer

$\frac{6}{7}, \frac{-3}{7}, \frac{2}{7}$

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(A) Let the vector be $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$.Given that the projections of the vector on the coordinate axes are $6, -3, 2$,we have $a = 6$,$b = -3$,and $c = 2$.The magnitude of the vector is $|\vec{r}| = \sqrt{a^2 + b^2 + c^2} = \sqrt{6^2 + (-3)^2 + 2^2}$.$|\vec{r}| = \sqrt{36 + 9 + 4} = \sqrt{49} = 7$.The direction cosines $(l, m, n)$ of a vector are given by $\frac{a}{|\vec{r}|}, \frac{b}{|\vec{r}|}, \frac{c}{|\vec{r}|}$.Thus,$l = \frac{6}{7}$,$m = \frac{-3}{7}$,and $n = \frac{2}{7}$.Therefore,the direction cosines are $\frac{6}{7}, \frac{-3}{7}, \frac{2}{7}$.

The projections of a vector on the three coordinate axes are $6, -3, 2$ respectively. The direction cosines of the vector are . . . . . . .

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