Question

Show that the vector \(\hat{i}+\hat{j}+\hat{k}\) is equally inclined to the axes OX,OY,and OZ.

Answer 1

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 \(\vec{a}\) are\((\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}).\)
Now,let \(α,β\),and \(γ\) be the angles formed by \(\vec{a}\) with the positive directions \(x,y\),and \(z\) axes.
Then,we have \(cosα=\frac{1}{\sqrt{3}},cosβ=\frac{1}{\sqrt{3}},cosγ=\frac{1}{\sqrt{3}}.\)
Hence,the given vector is equally inclined to axes OX,OY,and OZ.