Q.Find the direction cosines of $x$, $y$ and $z$-axis.

Concept: Direction Cosines Properties — The direction cosines of a line are the cosines of the angles it makes with the positive $x$, $y$, and $z$-axes. For a coordinate axis, the line lies along that axis itself.

Step 1: For the $x$-axis, the angles it makes with the axes are $0^\circ$ with $x$, $90^\circ$ with $y$, and $90^\circ$ with $z$.

Step 2: Taking cosines: $\cos 0^\circ = 1$, $\cos 90^\circ = 0$, $\cos 90^\circ = 0$. So direction cosines are $(1, 0, 0)$. …

Why Direction Cosines? The Core Idea

Direction cosines are a way to describe the orientation of a line in 3D space. Instead of giving a vector, we give three numbers: the cosines of the angles the line makes with the positive $x$, $y$, and $z$ axes. These angles are usually denoted $\alpha$, $\beta$, and $\gamma$.

The key property is that for any line, the sum of the squares of its direction cosines is always 1:

where $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.

Now, what about the coordinate axes themselves? Each axis is just a special line. For the x-axis, it lies exactly along the $x$ direction. So the angle it makes with itself is $0^\circ$, and with the other two axes it's $90^\circ$. That's the entire geometric insight — the rest is just taking cosines.

Step-by-Step Solution

1. Direction cosines of the x-axis

The x-axis is the line through the origin pointing along the positive $x$ direction.

• Angle with x-axis: $\alpha = 0^\circ \implies \cos 0^\circ = 1$
• Angle with y-axis: $\beta = 90^\circ \implies \cos 90^\circ = 0$
• Angle with z-axis: $\gamma = 90^\circ \implies \cos 90^\circ = 0$

So the direction cosines are $(1, 0, 0)$.

2. Direction cosines of the y-axis

The y-axis points along the positive $y$ direction.

• Angle with x-axis: $\alpha = 90^\circ \implies \cos 90^\circ = 0$
• Angle with y-axis: $\beta = 0^\circ \implies \cos 0^\circ = 1$
• Angle with z-axis: $\gamma = 90^\circ \implies \cos 90^\circ = 0$

So the direction cosines are $(0, 1, 0)$.

3. Direction cosines of the z-axis

The z-axis points along the positive $z$ direction.

• Angle with x-axis: $\alpha = 90^\circ \implies \cos 90^\circ = 0$
• Angle with y-axis: $\beta = 90^\circ \implies \cos 90^\circ = 0$ …

Method: Direction Cosines of a Coordinate Axis

Use this whenever you need the direction cosines of one of the coordinate axes — or, more generally, of any line whose orientation relative to the axes you can read off directly.

Steps

Step 1: State the angles the axis makes with all three axes.

A coordinate axis makes $0^\circ$ with itself and $90^\circ$ with each of the other two axes. So the x-axis has angles $(0^\circ,90^\circ,90^\circ)$, and similarly for the others.

Step 2: Take cosines.

$l=\cos\alpha,\quad m=\cos\beta,\quad n=\cos\gamma,\qquad \cos0^\circ=1,\ \cos90^\circ=0$ …

Common Mistakes

Mistake 1: Writing the x-axis direction cosines as $(1,1,1)$.

Why it's wrong: an axis makes a $90^\circ$ angle with each of the other two axes, and $\cos 90^\circ=0$; only the cosine with itself is $1$. Correct approach: the x-axis gives $(1,0,0)$, the y-axis $(0,1,0)$, the z-axis $(0,0,1)$.

Mistake 2: Confusing which slot holds the $1$. …