Q.If a line makes angle $90^\circ$, $60^\circ$ and $30^\circ$ with the positive direction of $x$, $y$ and $z$-axis respectively, find its direction cosines.

The key idea is that direction cosines are the cosines of the angles a line makes with the positive coordinate axes.

Let the angles be $\alpha = 90^\circ$, $\beta = 60^\circ$, and $\gamma = 30^\circ$. The direction cosines are $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$.

Compute each:

• $l = \cos 90^\circ = 0$
• $m = \cos 60^\circ = \frac{1}{2}$
• $n = \cos 30^\circ = \frac{\sqrt{3}}{2}$

These satisfy the property $l^2 + m^2 + n^2 = 1$ (since $0 + \frac{1}{4} + \frac{3}{4} = 1$), confirming they are valid.

The Core Idea: What Direction Cosines Really Mean

Direction cosines are not just a formula — they are the coordinates of a unit vector pointing along the line. If a line makes angles $\alpha$, $\beta$, $\gamma$ with the positive $x$, $y$, $z$ axes, then its direction cosines are:

The key property that makes this concept powerful is that these three numbers always satisfy:

Why? Because the direction cosines are the components of a unit vector. This is your built-in sanity check — if the squares don't sum to 1, something is wrong.

Step-by-Step Solution

1. Identify the given angles

The line makes:

• $\alpha = 90^\circ$ with the $x$-axis
• $\beta = 60^\circ$ with the $y$-axis
• $\gamma = 30^\circ$ with the $z$-axis

2. Compute each direction cosine directly

3. Verify the fundamental property

Check that $l^2 + m^2 + n^2 = 1$:

This confirms our answer is consistent. If the sum had been anything other than 1, we would know an error crept in.

4. Write the direction cosines as an ordered triple

The direction cosines are $(l, m, n) = \left(0, \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Method: Direction Cosines from the Angles a Line Makes with the Axes

Use this when a line's angles $\alpha,\beta,\gamma$ with the positive $x$, $y$, $z$ axes are given (directly or indirectly) and you need its direction cosines.

Steps

Step 1: Take the cosine of each given angle.

The direction cosines are the cosines themselves — not the angles, and always of the angle actually given (do not swap in a complement).

Step 2: Verify with the fundamental identity.

If the squares do not add to $1$, an angle was misread or a cosine mis-evaluated. This same identity also lets you recover a third direction cosine when only two angles are supplied.

Step 3: Write them in order $(l,m,n)$.

Keep the $x,y,z$ order; the triple is only meaningful with each cosine attached to its own axis. Reversing the line's sense flips all three signs, giving the equally valid set $(-l,-m,-n)$.

Common Mistakes

Mistake 1: Using the complement of the given angle.

Why it's wrong: the direction cosine is the cosine of the angle the line actually makes with that axis — for $60^\circ$ with the y-axis it is $\cos 60^\circ=\tfrac12$, not $\cos 30^\circ$. Correct approach: take the cosine of each given angle directly.

Mistake 2: Reporting the angles themselves instead of their cosines.

Why it's wrong: direction cosines are numbers like $0,\tfrac12,\tfrac{\sqrt3}{2}$, not $90^\circ,60^\circ,30^\circ$. Correct approach: convert every angle to its cosine, then optionally verify $l^2+m^2+n^2=1$.