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Find the direction ratios and direction cosines of the vector \overrightarrow{\mathrm{a}}=(5 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}).
CBSE | Class 12 | Maths | RS Aggarwal | Vectors and Their Properties
Find the direction ratios and direction cosines of the vector \overrightarrow{\mathrm{a}}=(5 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}).

Solution:

\vec{a}=5 \hat{\imath}-3 \hat{\jmath}+4 \hat{k}
Direction ratios are ratios of \mathrm{x}, \mathrm{y} and \mathrm{z} component of the vector while direction cosine are cosines of angle between the vector and \mathrm{x}-axis, y-axis and z – axis respectively.

Also sum of square of direction cosines is 1 . In the given question direction ratios are (5,-3,4) while for direction cosines, we need to find the unit vector
\hat{a}=\frac{\vec{a}}{I \vec{a} \mathrm{I}}=\frac{5 \hat{\imath}-3 \hat{\jmath}+4 \hat{k}}{\sqrt{50}}
So in this question direction cosines are \left(\frac{5}{5 \sqrt{2}}, \frac{-3}{5 \sqrt{2}}, \frac{4}{5 \sqrt{2}}\right)

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