If a line has the direction ratios $-18, 12, -4$, then what are its direction cosines ?

Question

Accepted Answer

> [!TLDR]
> Direction cosines are the cosines of the angles a line makes with the axes, found by dividing each direction ratio by the magnitude of the direction vector. For ratios $-18, 12, -4$, the direction cosines are $\left( -\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} ight)$.

## Why Direction Ratios and Cosines?

A line in space can be described by a vector parallel to it. The components of that vector — say $(a, b, c)$ — are called **direction ratios**. They tell you the relative "step" the line takes along each axis, but they are not unique: scaling $(a, b, c)$ by any non-zero constant gives the same line.

Direction **cosines** are the unique, normalized version. They are the cosines of the angles $\alpha, \beta, \gamma$ that the line makes with the $x$, $y$, and $z$ axes respectively. If a direction vector is $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$, then:

> [!FORMULA]
> $$ \cos\alpha = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad \cos\beta = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad \cos\gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}} $$

The key property: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

So the task is simple: take the given ratios, compute the magnitude, and divide each ratio by it.

---

## Step-by-step

**1. Identify the direction ratios.**  
We are given $a = -18$, $b = 12$, $c = -4$.

**2. Compute the magnitude (the divisor).**  
The magnitude of the direction vector is:
$$ \sqrt{a^2 + b^2 + c^2} = \sqrt{(-18)^2 + 12^2 + (-4)^2} $$
$$ = \sqrt{324 + 144 + 16} = \sqrt{484} = 22 $$

> [!TIP]
> $484 = 4 	imes 121 = 22^2$, so the square root is exact. Always check if the sum of squares is a perfect square — it often simplifies nicely in exam problems.

**3. Divide each ratio by the magnitude.** …

Q.If a line has the direction ratios $-18, 12, -4$ , then what are its direction cosines ?

Concept understanding — Direction Cosines Properties

Direction Cosines and Their Properties

Property 1 — the squares sum to 1

Property 2 — they are a unit vector

Property 3 — fixed up to sign

Direction ratios

Why Direction Ratios and Cosines?

Step-by-step

Method: Normalise direction ratios to get direction cosines

Steps

Common Mistakes

Q.If a line has the direction ratios −18,12,−4-18, 12, -4−18,12,−4, then what are its direction cosines ?

Why Direction Ratios and Cosines?

Step-by-step

🎓Unlock everything free for 14 days

Q.If a line has the direction ratios $-18, 12, -4$ , then what are its direction cosines ?

Unlock everything free for 14 days