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 $O X, O Y$ and $O Z$ 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}$