Q.If a line makes angles $90^\circ$, $135^\circ$, $45^\circ$ with the $x, y$ and $z$-axes respectively, find its direction cosines.

Concept: Direction Cosines — the cosines of the angles a line makes with the coordinate axes.

The direction cosines are given directly by $\cos \theta_x$, $\cos \theta_y$, and $\cos \theta_z$.

• With the $x$-axis: $\theta_x = 90^\circ$, so $\cos 90^\circ = 0$.
• With the $y$-axis: $\theta_y = 135^\circ$, so $\cos 135^\circ = -\frac{1}{\sqrt{2}}$.
• With the $z$-axis: $\theta_z = 45^\circ$, so $\cos 45^\circ = \frac{1}{\sqrt{2}}$.

These three values satisfy the identity $\cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1$, confirming they are valid direction cosines.

Why Direction Cosines?

Direction cosines are the simplest way to describe a line’s orientation in 3D space. If a line makes an angle $\alpha$ with the $x$-axis, $\beta$ with the $y$-axis, and $\gamma$ with the $z$-axis, then the numbers $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are called its direction cosines. They are often denoted by $l$, $m$, $n$.

The key property: for any line, $l^2 + m^2 + n^2 = 1$. This is because the direction cosines are the components of a unit vector along the line.

Here, the angles are given directly — so we just need to compute the cosines. No extra work is needed.

Step-by-step

1. Identify the angles.

   The line makes:

   • $\alpha = 90^\circ$ with the $x$-axis,
   • $\beta = 135^\circ$ with the $y$-axis,
   • $\gamma = 45^\circ$ with the $z$-axis.

2. Compute each direction cosine.

   • $l = \cos 90^\circ = 0$
   • $m = \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{1}{\sqrt{2}}$
   • $n = \cos 45^\circ = \frac{1}{\sqrt{2}}$

3. Verify the fundamental relation.

   Check: $l^2 + m^2 + n^2 = 0^2 + \left(-\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 0 + \frac{1}{2} + \frac{1}{2} = 1$.

   This confirms the numbers are valid direction cosines.

Method: Direction cosines straight from the axis angles

Use this when a line's angles with the three coordinate axes are given and you need its direction cosines.

Steps

Step 1: Recall the definition.

If a line makes angles $\alpha,\beta,\gamma$ with the $x,y,z$ axes, its direction cosines are

The answer is the COSINES of the angles, never the angles themselves.

Step 2: Evaluate each cosine, watching the quadrant sign.

An angle above $90^\circ$ gives a negative cosine (e.g. $\cos135^\circ=-\cos45^\circ=-\tfrac{1}{\sqrt2}$). Use $\cos(180^\circ-\theta)=-\cos\theta$ if unsure.

Step 3: Verify with the identity.

Check that $l^2+m^2+n^2=1$. If it fails, a cosine value or a sign is wrong.

Common Mistakes

Mistake 1: Making $\cos135^\circ$ positive.

Why it's wrong: $135^\circ$ lies in the second quadrant where cosine is negative, so $\cos135^\circ=-\frac{1}{\sqrt2}$, not $+\frac{1}{\sqrt2}$. Correct approach: use $\cos(180^\circ-\theta)=-\cos\theta$.

Mistake 2: Reporting the angles themselves as the direction cosines.

Why it's wrong: direction cosines are the COSINES of the angles, so the answer is $0,-\frac{1}{\sqrt2},\frac{1}{\sqrt2}$, not $90^\circ,135^\circ,45^\circ$. Correct approach: take the cosine of each angle and confirm $l^2+m^2+n^2=1$.