Question

If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines.

Answer 1

Let us assume the angles that made with the positive direction of x, y, and z-axes be α, β, γ.

Then we get

\(\Rightarrow \alpha = 90^\circ\)

\(\Rightarrow \beta = 60^\circ\)

\(\Rightarrow \gamma = 30^\circ\)

We know that if a line makes angles of α, β, γ with the positive x, y, and z-axes then the direction cosines of that line is the cosine of that angles made by that line with the axes.

Let us assume that l, m, n are the direction cosines of the line. Then

\(\Rightarrow \) l = cos α

\(\Rightarrow \) m = cos β

\(\Rightarrow \) n = cos γ

We substitute the values of α, β, γ in the above equations for the values of l, m, n.

 \(\Rightarrow \) l = cos\((90^\circ)\)

 \(\Rightarrow \) l = 0

 \(\Rightarrow \) m =  cos\((60^\circ)\)

 \(\Rightarrow \) m = \(\frac{1}{2}\)

 \(\Rightarrow \) n =  cos\((30^\circ)\)

 \(\Rightarrow \) n = \(\frac{\sqrt{3}}{2}\)

∴ The direction cosines of the given line is 0, \(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\)

Answer 2

Direction cosines are the cosines of the angles that a line makes with the axes.

So, the direction cosines are Cos90^o, Cos60^o and Cos30^o = 0, 1/2, √3/2

Hope it helps

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