Find the direction cosines of the vector î + 2ĵ - k̂.

Concept:

The direction cosines of the vector aî + bĵ + ck̂ are given by α = \(\rm \frac{a}{\sqrt{a^2+b^2+c^2}}\), β = \(\rm \frac{b}{\sqrt{a^2+b^2+c^2}}\) and γ = \(\rm \frac{c}{\sqrt{a^2+b^2+c^2}}\).

Calculation:

For the given vector î + 2ĵ - k̂, a = 1, b = 2 and  = -1.

The direction cosines of the vector are:

α = \(\rm \frac{1}{\sqrt{1^2+2^2+(-1)^2}}\), β = \(\rm \frac{2}{\sqrt{1^2+2^2+(-1)^2}}\) and γ = \(\rm \frac{-1}{\sqrt{1^2+2^2+(-1)^2}}\)

⇒ α = \(\rm \frac{1}{\sqrt{6}}\), β = \(\rm \frac{2}{\sqrt{6}}\) and γ = \(\rm \frac{-1}{\sqrt{6}}\)