Question

If $\alpha ,\beta$ and $\gamma$ are the direction angles of vector $a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$. Based on above information, which of the following is incorrect?

• A. Direction cosines are $\cos \alpha =\frac{a_1}{|a|},\cos \beta =\frac{a_2}{|a|}$ and $\cos \gamma =\frac{a_3}{|a|}$
• B. The scalar components $a_1,a_2$ and $a_3$ of the vector $a$, are precisely the projections of $a$ along $X,Y$ and $Z$-axes, respectively
• C. If a is a unit vector, then $a=\frac{\hat{i}}{\cos \alpha }+\frac{\hat{j}}{\cos \beta }+\frac{\hat{k}}{\cos \gamma }$
• D. Projection of $a$ along $OX,OY$ and $OZ$ are $|a|\cos \alpha ,|a|\cos \beta$, $|a|\cos \gamma$ respectively

Answer 1

Answer: (C)

If $\alpha ,\beta \gamma$ are the direction angles of vector $a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, then its direction cosines are
$\cos \alpha =\frac{a\cdot \hat{i}}{|a||\hat{i}\mid }=\frac{a_1}{|a|}$ similarly, $\cos \beta =\frac{a_2}{|a|}$ and $\cos \gamma =\frac{a_3}{|a|}$
Since, a is a unit vector
$\cos \alpha =a_1,\cos \beta =a_2,\cos \alpha =a_3$
Thus, $a=\cos \alpha \hat{i}+\cos \beta \hat{j}+\cos \gamma \hat{k}$