Q.Find the direction cosines of a line which makes equal angles with the coordinate axes.

The key idea is that if a line makes equal angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes, then $\alpha = \beta = \gamma$.

Since direction cosines satisfy $l^2 + m^2 + n^2 = 1$, and here $l = m = n = \cos \alpha$, we have:

Thus each direction cosine is $\pm \frac{1}{\sqrt{3}}$, and all three must have the same sign (all positive or all negative) for a consistent direction.

Why Direction Cosines?

Direction cosines are the cosines of the angles a line makes with the positive $x$, $y$, and $z$ axes. If a line makes angles $\alpha$, $\beta$, and $\gamma$ with the axes, then its direction cosines are $l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$.

The key property: for any line in 3D space, the sum of the squares of its direction cosines is always exactly 1:

This is not a coincidence — it follows from the fact that a unit vector along the line has components $(l, m, n)$, and its magnitude must be 1.

Step-by-step solution

1. Set up the equal-angle condition

The problem says the line makes equal angles with all three coordinate axes. That means:

Taking cosines of both sides of each equality:

So all three direction cosines are equal:

2. Apply the fundamental relation

Substitute $l = m = n$ into $l^2 + m^2 + n^2 = 1$:

3. Solve for the direction cosines

Taking square roots:

Since $l = m = n$, all three are the same:

4. Write the final answer

The direction cosines are either all positive or all negative:

Method: Use $l^2+m^2+n^2=1$ to pin down constrained direction cosines

When a line is described by a condition on the angles it makes with the axes (equal angles, a stated angle with one axis, ...) rather than by explicit numbers, translate the words into relations among $l,m,n$ and close the system with the identity $l^2+m^2+n^2=1$.

Steps

Step 1: Name the direction cosines. Let $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$ be the cosines of the angles the line makes with the $x,y,z$ axes.

Step 2: Encode the given condition. "Equal angles with the axes" means $\alpha=\beta=\gamma$, hence $l=m=n$. (A different phrasing — a known angle with one axis — would instead fix one of $l,m,n$ directly.)

Step 3: Apply the identity. Substitute the relation into

With $l=m=n$ this becomes $3l^2=1$, so $l=\pm\dfrac{1}{\sqrt3}$.

Step 4: Keep the sign choice consistent. Reversing a line flips all three cosines together, so the valid answers are the whole triple positive or the whole triple negative — never a mix. Report both.

The identity does the heavy lifting: it converts one algebraic constraint into numerical values, which is why this same routine solves every "find the direction cosines from a condition" problem.

Common Mistakes

Mistake 1: Using $l+m+n=1$ instead of $l^2+m^2+n^2=1$.

Why it's wrong: only the sum of squares of direction cosines equals $1$. Correct approach: with $l=m=n$, solve $3l^2=1$, giving $l=\pm\frac{1}{\sqrt3}$ — not $3l=1$.

Mistake 2: Dropping the $\pm$ and reporting only the positive triple.

Why it's wrong: a line has two opposite senses, so both $\left(\frac{1}{\sqrt3},\frac{1}{\sqrt3},\frac{1}{\sqrt3}\right)$ and its negative are valid. Correct approach: give both sign choices.

Mistake 3: Mixing signs, e.g. $\left(\frac{1}{\sqrt3},-\frac{1}{\sqrt3},\frac{1}{\sqrt3}\right)$.

Why it's wrong: equal angles force $l=m=n$ exactly, so the three cosines must share one sign. Correct approach: flip all three together, or none.