Find the direction ratios and the direction cosines of the vector joining the points $A(2,1,-2)$ and $B(3,5,-4)$.

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Find the direction ratios and the direction cosines of the vector joining the points $A(2,1,-2)$ and $B(3,5,-4)$ .

CBSE | Class 12 | Maths | RS Aggarwal | Vectors and Their Properties

Find the direction ratios and the direction cosines of the vector joining the points $A(2,1,-2)$ and $B(3,5,-4)$ .

Solution:

$\begin{array}{l} \overrightarrow{O A}=2 \hat{\imath}+\hat{\jmath}-2 \hat{k} \\ \overrightarrow{O B}=3 \hat{\imath}+5 \hat{\jmath}-4 \hat{k} \\ \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=\hat{\imath}+4 \hat{\jmath}-2 \hat{k} \end{array}$
Clearly direction ratios of $\overrightarrow{A B}$ are $(1,4,-2)$
For direction cosines we need to find direction i.e. unit vector $\widehat{A B}=\frac{\overrightarrow{A B}}{I \overrightarrow{A B} I}=\frac{\hat{\imath}+4 \hat{\jmath}-2 \hat{k}}{\sqrt{1^{2}+4^{2}+2^{2}}}=\frac{\hat{\imath}+4 \hat{\jmath}-2 \hat{k}}{\sqrt{21}}$
Direction cosine $\operatorname{are}\left(\frac{1}{\sqrt{21}}, \frac{4}{\sqrt{21}}, \frac{-2}{\sqrt{21}}\right)$
Note: While writing direction ratios or cosines, the order matters.