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Q.
The direction cosines of the vector $\hat{ i }+2 \hat{ j }+3 \hat{ k }$ are
Vector Algebra
Solution:
To find the direction of a given vector, we have to find the unit vector in the direction of given vector.
Let $ a=\hat{i}+2 \hat{j}+3 \hat{k} $
Then, $ |a|=\sqrt{1^2+2^2+3^2}=\sqrt{14}$
$ \therefore \hat{a}=\frac{a}{|a|} \Rightarrow \hat{a}=\frac{1}{\sqrt{14}}(\hat{i}+2 \hat{j}+3 \hat{k})$
$ \Rightarrow \hat{a}=\frac{1}{\sqrt{14}} \hat{i}+\frac{2}{\sqrt{14}} \hat{j}+\frac{3}{\sqrt{14}} \hat{k}$
Hence, direction cosines of the given vector are $\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}$
$(\because$ direction cosines are the coefficients of $\hat{i}, \hat{j}, \hat{k}$ of unit vector).