Direction cosines and direction ratios of a line joining two points

Question

Find the direction cosines and direction ratios of the line joining the points $A(2, 1, -3)$ and $B(4, -1, 5)$ .

(NCERT Class 12, Chapter 11 — Three Dimensional Geometry)

Solution — Step by Step

Direction ratios (DRs) of the line $AB$ are simply the differences in coordinates:

a = x_2 - x_1 = 4 - 2 = 2

b = y_2 - y_1 = -1 - 1 = -2

c = z_2 - z_1 = 5 - (-3) = 8

Direction ratios: (2, -2, 8) or equivalently (1, -1, 4) (dividing by 2).

|AB| = \sqrt{a^2 + b^2 + c^2} = \sqrt{4 + 4 + 64} = \sqrt{72} = 6\sqrt{2}

Direction cosines (DCs) are the direction ratios divided by the magnitude:

l = \frac{a}{|AB|} = \frac{2}{6\sqrt{2}} = \frac{1}{3\sqrt{2}}

m = \frac{b}{|AB|} = \frac{-2}{6\sqrt{2}} = \frac{-1}{3\sqrt{2}}

n = \frac{c}{|AB|} = \frac{8}{6\sqrt{2}} = \frac{4}{3\sqrt{2}}

Direction cosines: $\left(\frac{1}{3\sqrt{2}}, \frac{-1}{3\sqrt{2}}, \frac{4}{3\sqrt{2}}\right)$

Verification: $l^2 + m^2 + n^2 = \frac{1}{18} + \frac{1}{18} + \frac{16}{18} = \frac{18}{18} = 1$ ✓

Why This Works

Direction ratios tell us the “direction” of a line — they’re proportional to the components of the line’s direction vector. Any scalar multiple of DRs gives the same direction, so $(2, -2, 8)$ and $(1, -1, 4)$ represent the same line direction.

Direction cosines are the normalised version — they’re the cosines of the angles the line makes with the three coordinate axes ( $x$ , $y$ , $z$ ). Because they represent cosines of angles in a unit direction, they always satisfy $l^2 + m^2 + n^2 = 1$ . This property is a useful self-check.

The relationship is: DCs = DRs $\div$ magnitude. DCs are unique (up to sign), while DRs can be any proportional set.

Alternative Method — Using the direction vector directly

The direction vector of $AB$ is $\vec{AB} = B - A = (2, -2, 8)$ .

The unit vector: $\hat{AB} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{(2, -2, 8)}{6\sqrt{2}} = \left(\frac{1}{3\sqrt{2}}, \frac{-1}{3\sqrt{2}}, \frac{4}{3\sqrt{2}}\right)$

The components of the unit vector ARE the direction cosines.

Always verify $l^2 + m^2 + n^2 = 1$ after computing direction cosines. If it doesn’t equal 1, you’ve made an arithmetic error somewhere. This check takes 10 seconds and can save you from a wrong answer. For CBSE boards, showing this verification step also earns marks.

Common Mistake

Students confuse direction ratios with direction cosines. Direction ratios are NOT unique — $(2, -2, 8)$ , $(1, -1, 4)$ , and $(4, -4, 16)$ all represent the same direction. Direction cosines ARE unique (up to a $\pm$ sign). When the question asks for “direction cosines,” you MUST divide by the magnitude. When it asks for “direction ratios,” any proportional set is acceptable.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the direction vector directly

Common Mistake

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