Q.Write the direction ratios of the vector $\vec{a}=\hat{i}+\hat{j}-2\hat{k}$ and hence calculate its direction cosines.

Concept: Direction Cosines Properties – For a vector $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, direction ratios are $(a_1, a_2, a_3)$ and direction cosines are $\left( \frac{a_1}{|\vec{a}|}, \frac{a_2}{|\vec{a}|}, \frac{a_3}{|\vec{a}|} \right)$.

Step 1: Identify direction ratios.

For $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$, the direction ratios are $(1, 1, -2)$.

Step 2: Compute magnitude.

$|\vec{a}| = \sqrt{1^2 + 1^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$.

Step 3: Divide each direction ratio by $|\vec{a}|$ to get direction cosines. …

Concept First: What Are Direction Ratios and Direction Cosines?

Any vector in space can be described by its components along the $x$, $y$, and $z$ axes. These components are called the direction ratios (or direction numbers) of the vector. They tell you how much the vector moves in each direction.

But a vector's direction is independent of its length. If you scale a vector, its direction stays the same. So to talk purely about direction, we use direction cosines — the cosines of the angles the vector makes with the positive $x$, $y$, and $z$ axes. These are just the direction ratios divided by the vector's magnitude.

The key property: the sum of squares of direction cosines always equals 1. This is because they represent the components of a unit vector in the same direction.

Step-by-Step Solution

1. Identify the direction ratios.

The vector is $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$. The coefficients of $\hat{i}$, $\hat{j}$, and $\hat{k}$ are $1$, $1$, and $-2$ respectively.

So the direction ratios are $(1, 1, -2)$.

2. Calculate the magnitude of the vector.

The magnitude is the square root of the sum of squares of the direction ratios: …

Method: Direction ratios and direction cosines of a vector

Use this to describe the direction of a vector by the angles it makes with the coordinate axes.

Steps

Step 1: Read off the direction ratios.

For $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, the direction ratios are simply the components $(a_1,a_2,a_3)$ — signs included.

Step 2: Divide by the magnitude to get direction cosines.

$|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad l=\frac{a_1}{|\vec{a}|},\ m=\frac{a_2}{|\vec{a}|},\ n=\frac{a_3}{|\vec{a}|}.$ …

Common Mistakes

Mistake 1: Dropping the sign of the $-2$ direction ratio/cosine.

Why it's wrong: the direction ratios are $(1,1,-2)$ and the cosine along $z$ is $-\frac{2}{\sqrt6}$; the sign encodes that the vector points in the negative-$z$ sense. Correct approach: carry every component's sign through into both ratios and cosines.

Mistake 2: Reporting the angles instead of their cosines (or writing $l+m+n=1$). …