Show that the direction cosines of a vector equally inclined to the axes `O X ,\ O Y\ a n d\ O Z` are `1/(sqrt(3)),\ 1/(sqrt(3)),1/(sqrt(3))dot`

Explanation

Let `vec{a}=hat{i}+hat{j}+hat{k}.`

Then,

`|vec{a}|=sqrt{1^{2}+1^{2}+1^{2}}=sqrt{3}`

Therefore, the direction cosines of `quad vec{a} `
are `(frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}}).`

Now, let` alpha, beta, and gamma` be the angles formed by with the positive directions of` x, y, and z `axes.

Then, we have


`cos alpha=frac{1}{sqrt{3}}, cos beta=frac{1}{sqrt{3}}, cos gamma=frac{1}{sqrt{3}} .`

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