Find the angles which the vector $\vec{a}=3 \hat{i}-6 \hat{j}+2 \hat{k}$ makes with the coordinate axes.

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CBSE | Class 12 | Maths | RS Aggarwal | Scalar, or Dot, Product of Vectors

Find the angles which the vector $\vec{a}=3 \hat{i}-6 \hat{j}+2 \hat{k}$ makes with the coordinate axes.

Solution:

Direction cosines gives the angle made by the vector with $x$ -axis, $y-$ axis, z-axis respectively.
$\begin{array}{l} \vec{a}=3 \hat{\imath}-6 \hat{\jmath}+2 \hat{k} \\ \mathrm{I} \vec{a} \mathrm{I}=\sqrt{3^{2}+6^{2}+2^{2}}=\sqrt{9+36+4}=7 \end{array}$
To find direction cosines, its sufficient to find unit vector.
$\hat{a}=\frac{\vec{a}}{\mathrm{I} \vec{a} \mathrm{I}}=\frac{3 \hat{\imath}-6 \hat{\jmath}+2 \hat{k}}{7}=\frac{3}{7} \hat{\imath}-\frac{6}{7} \hat{\jmath}+\frac{2}{7} \hat{k}$
Direction cosines are $\left(\frac{3}{7},-\frac{6}{7}, \frac{2}{7}\right)$
Angle made with $x$ -axis is $\cos \alpha=\frac{3}{7}$
$\alpha=\cos ^{-1} \frac{3}{7}$
Angle made with $y$ -axis is $\cos \beta=-\frac{6}{7}$
$\beta=\cos ^{-1} \frac{-6}{7}$
Angle made with $z$ -axis is $\cos \gamma=\frac{2}{7}$
$\gamma=\cos ^{-1} \frac{2}{7}$