Q.Find the direction cosines of the line passing through the two points $(-2, 4, -5)$ and $(1, 2, 3)$.

Concept: Direction Cosines of a Line — the cosines of the angles a line makes with the coordinate axes, equal to the components of the unit vector along the line.

Step 1: Find the direction ratios.

For points $(-2,4,-5)$ and $(1,2,3)$, the direction ratios are:

$(1 - (-2),\; 2 - 4,\; 3 - (-5)) = (3,\; -2,\; 8)$.

Step 2: Compute the magnitude.

$|\vec{AB}| = \sqrt{3^2 + (-2)^2 + 8^2} = \sqrt{9 + 4 + 64} = \sqrt{77}$. …

Why Direction Cosines? The Core Idea

A line in 3D space doesn't have a unique "starting point" — it's defined by its direction. Direction cosines capture that direction in a pure, unit-free way. They are the cosines of the three angles the line makes with the positive $x$, $y$, and $z$ axes. Because they come from a unit vector along the line, they always satisfy the beautiful relation:

where $l, m, n$ are the direction cosines. This is the Pythagorean theorem in 3D for a vector of length 1.

The trick: we first find the direction ratios (any numbers proportional to the direction cosines) by subtracting coordinates. Then we normalise them to get the actual cosines.

Step-by-Step Solution

1. Find the direction ratios of the line.

The direction ratios (DRs) are simply the differences in the coordinates of the two given points. If a line passes through $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the DRs are $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$.

Here, $A = (-2, 4, -5)$ and $B = (1, 2, 3)$.

So:

• $x$-difference: $1 - (-2) = 3$
• $y$-difference: $2 - 4 = -2$
• $z$-difference: $3 - (-5) = 8$

Thus the direction ratios are $(3, -2, 8)$.

2. Compute the magnitude (length) of this direction vector. …

Method: Direction Cosines of the Line Through Two Points

Use this when a line is given by two points and you need its direction cosines (or direction ratios).

Steps

Step 1: Subtract coordinates to get direction ratios.

For $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, the direction ratios are $(x_2-x_1,\;y_2-y_1,\;z_2-z_1)$. Subtract consistently in one order (do $B-A$ throughout); doing $A-B$ instead simply reverses all signs and describes the same line.

Step 2: Normalise by the magnitude. …

Common Mistakes

Mistake 1: Sign error when subtracting a negative coordinate.

Why it's wrong: the $z$-ratio is $3-(-5)=8$, not $3-5=-2$; mishandling the double negative corrupts the direction ratios. Correct approach: subtract $B-A$ carefully to get $(3,-2,8)$.

Mistake 2: Forgetting to normalise, or normalising by the wrong length. …