Direction Cosines & Direction Ratios

MATDirection Cosines & Ratios

Direction cosines $l, m, n$ are the cosines of the angles a directed line makes with the positive $x$ -, $y$ - and $z$ -axes, while direction ratios $a, b, c$ are any numbers proportional to them. They are the basic language of 3D geometry, since every line, plane normal, and angle is described through them.

ISC examines converting ratios to cosines via normalisation and using the identity $l^2 + m^2 + n^2 = 1$ .

Fundamental identity of direction cosines

l^2 + m^2 + n^2 = 1

l = \cos\alpha,\ m = \cos\beta,\ n = \cos\gamma

where

\alpha, \beta, \gamma

are the angles the line makes with the positive

x

y

z

-axes.

Direction cosines from direction ratios

l = \dfrac{a}{\sqrt{a^2+b^2+c^2}},\quad m = \dfrac{b}{\sqrt{a^2+b^2+c^2}},\quad n = \dfrac{c}{\sqrt{a^2+b^2+c^2}}

(a,b,c)

are direction ratios; taking the

\pm

sign of the square root gives the two opposite directions of the line.

Direction ratios from two points

(a,b,c) = (x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1)

Line joining

A(x_1,y_1,z_1)

and

B(x_2,y_2,z_2)

; divide by

|AB| = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

to get direction cosines.

Angle between two lines via direction cosines

\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2

(l_1,m_1,n_1),(l_2,m_2,n_2)

are the direction cosines of the two lines;

\theta

is the angle between them.

Direction ratios are NOT unique — any non-zero scalar multiple $(ka, kb, kc)$ gives the same line; direction cosines are unique only up to an overall sign $\pm$ .
To convert ratios to cosines: compute the normalising factor $\sqrt{a^2+b^2+c^2}$ , then divide each ratio by it.
A line makes an angle of $90^\circ$ with an axis exactly when the corresponding direction cosine is $0$ (e.g. a line lying in the $yz$ -plane has $l = 0$ ).
If a line makes angles $\alpha, \beta, \gamma$ with the axes, then $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ and hence $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$ .
Direction cosines of the coordinate axes are: $x$ -axis $(1,0,0)$ , $y$ -axis $(0,1,0)$ , $z$ -axis $(0,0,1)$ .
A line along the vector $\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}$ has $a, b, c$ as its direction ratios automatically.
Always verify the answer by checking $l^2 + m^2 + n^2 = 1$ .

Where the marks go

Confusing direction ratios with direction cosines — ratios need NOT satisfy $a^2+b^2+c^2 = 1$ , only cosines do; you must normalise first.
Forgetting the $\pm$ ambiguity: a line has two sets of direction cosines (opposite senses). State both, or fix the sense if the question specifies it (e.g. directed from $A$ to $B$ ).
Sign errors when subtracting coordinates for direction ratios — always take $(x_2 - x_1)$ in a consistent order, never mixed.
When only two angles (say with the $x$ - and $y$ -axes) are given, students forget to use $l^2+m^2+n^2=1$ to find the third cosine, and may drop the valid second value.

How the board asks it

Numericaldirection cosines from direction ratios
Find the direction cosines of the line whose direction ratios are $2, -3, 6$ , and verify that $l^2 + m^2 + n^2 = 1$ .
Numericaldirection ratios from two points
Find the direction cosines of the line joining the points $A(1, 2, -3)$ and $B(4, 5, 1)$ , directed from $A$ to $B$ .
Numericalthe identity $l^2+m^2+n^2=1$
A line makes angles $60^\circ$ and $45^\circ$ with the positive $x$ - and $y$ -axes respectively. Find the angle $\gamma$ it makes with the $z$ -axis.
Give reasonsratios vs cosines and the $\pm$ ambiguity
Can the numbers $1, 2, 2$ be the direction cosines of a line? Give reasons, and hence write down the direction cosines of the line whose direction ratios are $1, 2, 2$ .
Derive / prove $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, prove that $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$ .
Applicationangle between two lines via direction cosines
Show that the line through the points $(1, -1, 2)$ and $(3, 4, -2)$ is perpendicular to the line through the points $(0, 3, 2)$ and $(3, 5, 6)$ , using their direction cosines.

Practise this topic

NextEquation of a Line (Vector & Cartesian)

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.