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Chapter 11 · Three Dimensional Geometry

Example 2: Direction cosines from the direction ratios 2,−1,−22, -1, -2

ExamplesDirection ratios to direction cosines · Section 11.2
The question

If a line has direction ratios 2,−1,−22, -1, -2⁠, determine its direction cosines.

Exam answer

Here a=2a = 2⁠, b=−1b = -1⁠, c=−2c = -2⁠.

a2+b2+c2=22+(−1)2+(−2)2=4+1+4=3\begin{aligned} &\sqrt{a^2 + b^2 + c^2} \\ &= \sqrt{2^2 + (-1)^2 + (-2)^2} \\ &= \sqrt{4 + 1 + 4} = 3 \end{aligned}

Direction cosines: l=±aa2+b2+c2l = \pm\frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, and similarly mm⁠, nn⁠, with one sign for all three. Taking the ++ sign,

l=23,m=−13,n=−23l = \frac23,\quad m = \frac{-1}{3},\quad n = \frac{-2}{3}

∴\therefore The direction cosines are 23,−13,−23\dfrac23, -\dfrac13, -\dfrac23⁠.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

Direction ratios are the direction cosines multiplied by some nonzero number. To undo that, divide each ratio by the length a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠: this scaling, or the same with a minus sign, is the only one that makes the squares add up to 1.

Given, to find, and the plan
Given
Direction ratios a=2a = 2⁠, b=−1b = -1⁠, c=−2c = -2 of a line.
To find
Its direction cosines l,m,nl, m, n⁠.
Plan
(1) Write l=kal = ka⁠, m=kbm = kb⁠, n=kcn = kc⁠. (2) Use l2+m2+n2=1l^2 + m^2 + n^2 = 1 to find kk⁠. (3) Compute a2+b2+c2\sqrt{a^2 + b^2 + c^2} and divide. (4) Decide the sign. (5) Check.

Basics you need

1Direction ratios

Any three numbers a,b,ca, b, c proportional to the direction cosines l,m,nl, m, n are direction ratios of the line: a=λla = \lambda l⁠, b=λmb = \lambda m⁠, c=λnc = \lambda n for some nonzero λ\lambda (Section 11.2). Basics: direction cosines

2The squares of the direction cosines add up to 1

l2+m2+n2=1l^2 + m^2 + n^2 = 1⁠, because li^+mj^+nk^l\hat i + m\hat j + n\hat k is a unit vector. Basics: Pythagoras in 3D

3Two directions on one line

Reversing a line changes the sign of all three direction cosines, so a line has two sets of direction cosines, l,m,nl, m, n and −l,−m,−n-l, -m, -n⁠. Lesson: direction cosines and direction ratios

Solution

  1. Write the direction cosines in terms of the ratios

    Since l,m,nl, m, n are proportional to 2,−1,−22, -1, -2⁠,

    l2=m−1=n−2=k,sol=2k, m=−k, n=−2k.\begin{aligned} &\frac{l}{2} = \frac{m}{-1} = \frac{n}{-2} = k, \\ &\text{so}\quad l = 2k,\ m = -k,\ n = -2k. \end{aligned}
    "Proportional" means each direction cosine is the same multiple kk of its direction ratio. One unknown number kk is left to find, and one condition is still unused: l2+m2+n2=1l^2 + m^2 + n^2 = 1⁠.
    Section 11.2: la=mb=nc=k\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k⁠. Basics: direction cosines

  2. Find kk from l2+m2+n2=1l^2 + m^2 + n^2 = 1

    (2k)2+(−k)2+(−2k)2=1k2(4+1+4)=19k2=1k=±13.\begin{aligned} (2k)^2 + (-k)^2 + (-2k)^2 &= 1 \\ k^2(4 + 1 + 4) &= 1 \\ 9k^2 &= 1 \\ k &= \pm\frac13. \end{aligned}
    This is the book's general derivation with numbers in it: k2(a2+b2+c2)=1k^2(a^2 + b^2 + c^2) = 1 gives k=±1a2+b2+c2k = \pm\frac{1}{\sqrt{a^2 + b^2 + c^2}}⁠. The square root of a2+b2+c2=9a^2 + b^2 + c^2 = 9 is 3, the length of the vector 2i^−j^−2k^2\hat i - \hat j - 2\hat k⁠. The equation k2=19k^2 = \frac19 has two roots, and both are allowed.
    k=±1a2+b2+c2k = \pm\frac{1}{\sqrt{a^2 + b^2 + c^2}} (Section 11.2). Basics: length in 3D

  3. Divide each ratio by 3

    With k=13k = \frac13⁠:

    l=23,m=−13,n=−23.l = \frac{2}{3}, \quad m = \frac{-1}{3}, \quad n = \frac{-2}{3}.
    These are the book's three fractions 222+(−1)2+(−2)2\frac{2}{\sqrt{2^2 + (-1)^2 + (-2)^2}}⁠, −1⋯\frac{-1}{\sqrt{\cdots}}⁠, −2⋯\frac{-2}{\sqrt{\cdots}}⁠, with the root worked out. Each ratio keeps its own sign: dividing by the positive number 3 does not change any sign.
    l=aa2+b2+c2l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, and the same for mm and nn⁠. Lesson: from ratios to cosines

  4. Decide the sign

    The other root, k=−13k = -\frac13⁠, gives −23,13,23-\frac23, \frac13, \frac23⁠.

    Both sets are direction cosines of the same line: one for each of its two directions. The ratios 2,−1,−22, -1, -2 point one way along the line, and the book takes the sign of kk that keeps that way, k=+13k = +\frac13⁠. The question does not fix a direction, so −23,13,23-\frac23, \frac13, \frac23 is also a correct set, but 23,−13,−23\frac23, -\frac13, -\frac23 is the expected answer.
    "Depending on the desired sign of kk⁠, either a positive or a negative sign is to be taken for l,ml, m and nn" (Section 11.2). Basics: reversing a line

  5. Check

    (23)2+(−13)2+(−23)2=49+19+49=1,\begin{aligned} &\Big(\frac23\Big)^2 + \Big(-\frac13\Big)^2 + \Big(-\frac23\Big)^2 \\ &= \frac49 + \frac19 + \frac49 = 1, \end{aligned}

    and 23:−13:−23=2:−1:−2\frac23 : -\frac13 : -\frac23 = 2 : -1 : -2⁠, the given ratios.

    Both properties are needed: the squares must add up to 1 (so they are direction cosines at all), and the three numbers must still be in the given proportion (so they belong to this line).
    l2+m2+n2=1l^2 + m^2 + n^2 = 1⁠. Basics: Pythagoras twice
Result

The direction cosines are 23,−13,−23\frac23, -\frac13, -\frac23 (or −23,13,23-\frac23, \frac13, \frac23 for the opposite direction of the same line).

(Textbook answer: 23,−13,−23\frac23, \frac{-1}{3}, \frac{-2}{3}⁠.)

Check it with numbers

As decimals, l≈0.666667l \approx 0.666667⁠, m≈−0.333333m \approx -0.333333⁠, n≈−0.666667n \approx -0.666667⁠, and 0.444444+0.111111+0.444444≈10.444444 + 0.111111 + 0.444444 \approx 1⁠.

The direction angles are cos⁡−123≈48.19∘\cos^{-1}\frac23 \approx 48.19^\circ⁠, cos⁡−1(−13)≈109.47∘\cos^{-1}\big(-\frac13\big) \approx 109.47^\circ and cos⁡−1(−23)≈131.81∘\cos^{-1}\big(-\frac23\big) \approx 131.81^\circ⁠. The negative cosines give obtuse angles: the line points away from the positive yy and zz-axes.

Common mistakes
  1. Dividing by 9 instead of 3. The divisor is a2+b2+c2=9=3\sqrt{a^2 + b^2 + c^2} = \sqrt9 = 3⁠, not a2+b2+c2a^2 + b^2 + c^2⁠.
  2. Squaring away the signs. (−1)2=1(-1)^2 = 1 inside the root, but the numerators keep their signs: m=−13m = -\frac13⁠, not 13\frac13⁠.
  3. Mixing signs from the two roots, for example 23,13,−23\frac23, \frac13, -\frac23⁠. All three signs change together, or none does.
What to take away
  1. Direction ratios to direction cosines: divide each by a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠.
  2. The ±\pm gives the two directions of the same line; the book takes ++⁠.
  3. Check: squares add up to 1, and the proportion is unchanged.

Every fact used, and where it comes from

FactUsed inSource
Direction ratios are proportional to the direction cosinesStep 1Section 11.2 (Basics)
l2+m2+n2=1l^2 + m^2 + n^2 = 1Steps 2, 5Section 11.2 (Basics)
l=±aa2+b2+c2l = \pm\frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, and the same for mm⁠, nnSteps 2, 3Section 11.2 (Lesson)
Reversing a line changes the signs of l,m,nl, m, nStep 4Section 11.2
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