Find the direction cosines of the vector 7î + 4ĵ - 3k̂.

Concept:

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

Calculation:

For the given vector 7î + 4ĵ - 3k̂, a = 7, b = 4 and c = -3.

The direction cosines of the vector are:

α = \(\rm\pm \frac{7}{\sqrt{7^2+4^2+(-3)^2}}\), β = \(\rm \pm \frac{4}{\sqrt{7^2+4^2+(-3)^2}}\) and γ = \(\rm\pm \frac{-3}{\sqrt{7^2+4^2+(-3)^2}}\)

⇒ α = \(\rm \pm \frac{7}{\sqrt{74}}\), β = \(\rm \pm \frac{4}{\sqrt{74}}\) and γ = \(\rm \frac{\mp 3}{\sqrt{74}}\) 

∴ (α , β , γ ) = (\(\rm \frac{7}{\sqrt{74}}, \rm \frac{4}{\sqrt{74}},\rm \frac{-3}{\sqrt{74}}\)) or (\(\rm \frac{-7}{\sqrt{74}}, \rm \frac{-4}{\sqrt{74}},\rm \frac{3}{\sqrt{74}}\)​)