Direction Angles and Direction Cosines

For a non-zero vector $\bar a$, its direction angles $\alpha,\beta,\gamma\in[0,\pi]$ are the angles it

makes with the positive $X,Y,Z$ axes, and their cosines $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$ are its

direction cosines (d.c.'s). Since $\alpha$ is the angle between $\hat\imath$ and $\bar a=a_1\hat\imath+a_2\hat\jmath+a_3\hat k$, $\cos\alpha=\dfrac{\bar a\cdot\hat\imath}{|\bar a|}=\dfrac{a_1}{|\bar a|}$, and similarly $\cos\beta=a_2/|\bar a|,\ \cos\gamma=a_3/|\bar a|$. Squaring and adding these three,

and $\bar a=|\bar a|(l\hat\imath+m\hat\jmath+n\hat k)$, so $(l,m,n)$ are exactly the components of the unit

vector along $\bar a$. The axes' own direction cosines are $\hat\imath:(1,0,0),\ \hat\jmath:(0,1,0),\ \hat k:(0,0,1)$, and a line $LL'$ has direction cosines $(l,m,n)$ in one sense and $(-l,-m,-n)$ in the other (the

two directions along the same line).

Direction ratios. Any three numbers $a,b,c$ proportional to the direction cosines ($a=\lambda l,\ b=\lambda m,\ c=\lambda n$ for some $\lambda$) are called direction ratios (d.r.'s). Since $l^2+m^2+n^2=1$ forces

$\lambda^2(a^2+b^2+c^2)=1$, the direction cosines are recovered from direction ratios by

(matching signs throughout). A line has infinitely many direction-ratio triples (any scalar multiple works)

but only one unsigned set of direction cosines.

Worked examples.

• Direction ratios $4,-12,18$ give magnitude $\sqrt{16+144+324}=22$, so direction cosines $\left(\tfrac{2}{11},-\tfrac{6}{11},\tfrac{9}{11}\right)$.
• A line perpendicular to two given lines (with direction ratios $-1,2,2$ and $0,2,1$): writing the unknown direction cosines $l,m,n$, the two perpendicularity conditions $-l+2m+2n=0$ and $2m+n=0$ give two linear equations, solved together (and then normalised by $l^2+m^2+n^2=1$) for the direction ratios/cosines of the perpendicular line.
• For a line making equations $5l+m+3n=0$ and $5mn-2nl+6lm=0$: eliminate $m$ using the first (linear) relation, substitute into the second (quadratic) relation to get a homogeneous quadratic in $l,n$ alone, factor it to …