Q.If a line has direction ratios $2, -1, -2$, determine its direction cosines.

The key idea is that direction cosines are the direction ratios normalized by the length of the direction vector.

Step 1: The direction ratios are $a = 2$, $b = -1$, $c = -2$.

Step 2: Compute the magnitude:

Step 3: Divide each direction ratio by this magnitude to get the direction cosines:

Why This Works: Direction Ratios vs. Direction Cosines

Direction ratios are any three numbers proportional to the direction cosines. They tell you the relative orientation of a line, but not the actual unit vector. Direction cosines, on the other hand, are the cosines of the angles the line makes with the coordinate axes — and they always satisfy $l^2 + m^2 + n^2 = 1$.

So if you have direction ratios $(a, b, c)$, the actual direction cosines are simply these numbers divided by the length of the vector $(a, b, c)$. That length is $\sqrt{a^2 + b^2 + c^2}$.

Step-by-Step Solution

1. Identify the direction ratios.

   The given direction ratios are $a = 2$, $b = -1$, $c = -2$.

2. Compute the magnitude (norm) of this vector.

   The magnitude is:

3. Divide each direction ratio by the magnitude to get the direction cosines.

4. Verify the sum of squares equals 1 (a quick sanity check).

This confirms the result is correct.

Method: Converting Direction Ratios to Direction Cosines

Use this whenever you have direction ratios (any three numbers proportional to the direction) and need the actual direction cosines.

Steps

Step 1: Treat the ratios as an un-normalised direction vector.

With ratios $(a,b,c)$, the line points along the vector $(a,b,c)$ — but this generally is not a unit vector.

Step 2: Divide each ratio by the vector's magnitude.

Normalising to length $1$ turns the direction vector into the direction cosines. Every sign must be carried through unchanged — a negative ratio gives a negative cosine.

Step 3: Sanity-check with $l^2+m^2+n^2=1$.

If the squares sum to $1$, the normalisation is correct. Note there are two valid answers, $(l,m,n)$ and $(-l,-m,-n)$, for the two senses of the line.

Common Mistakes

Mistake 1: Reporting the direction ratios as the direction cosines.

Why it's wrong: $(2,-1,-2)$ are only proportional to the cosines; the cosines must come from a unit vector. Correct approach: divide each by the magnitude $\sqrt{4+1+4}=3$ to get $\left(\tfrac{2}{3},-\tfrac{1}{3},-\tfrac{2}{3}\right)$.

Mistake 2: Dropping the negative signs during normalisation.

Why it's wrong: a negative ratio must give a negative cosine; forcing all cosines positive describes a different direction. Correct approach: carry each sign through, and confirm $l^2+m^2+n^2=1$.