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Chapter 11 · Three Dimensional Geometry

Exercise 11.1 · Q3: Direction cosines from the direction ratios −18,12,−4-18, 12, -4

Exercise 11.1Direction cosines from direction ratios · Section 11.2
The question

If a line has the direction ratios −18,12,−4-18, 12, -4⁠, then what are its direction cosines?

Exam answer

Here a=−18a = -18⁠, b=12b = 12⁠, c=−4c = -4⁠.

a2+b2+c2=(−18)2+122+(−4)2=324+144+16=484=22\begin{aligned} &\sqrt{a^2 + b^2 + c^2} \\ &= \sqrt{(-18)^2 + 12^2 + (-4)^2} \\ &= \sqrt{324 + 144 + 16} \\ &= \sqrt{484} = 22 \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=−1822=−911m=1222=611n=−422=−211\begin{aligned} l &= \frac{-18}{22} = -\frac{9}{11} \\ m &= \frac{12}{22} = \frac{6}{11} \\ n &= \frac{-4}{22} = -\frac{2}{11} \end{aligned}

∴\therefore The direction cosines are −911,611,−211-\dfrac{9}{11}, \dfrac{6}{11}, -\dfrac{2}{11}⁠.

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

The one idea this problem is built on

Direction ratios only fix the direction up to a scale; direction cosines are a scaled copy whose squares add up to 1 (there are two such copies, one the negative of the other). So divide each direction ratio by a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠, the length of the vector ai^+bj^+ck^a\hat i + b\hat j + c\hat k⁠.

Given, to find, and the plan
Given
Direction ratios a=−18a = -18⁠, b=12b = 12⁠, c=−4c = -4⁠.
To find
The direction cosines l,m,nl, m, n⁠.
Plan
(1) Compute a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠. (2) Divide each ratio by it. (3) Say what the ±\pm sign means. (4) Check, and see a shortcut.

Basics you need

1From direction ratios to direction cosines

If a,b,ca, b, c are direction ratios, then l=±aa2+b2+c2l = \pm\frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, m=±ba2+b2+c2m = \pm\frac{b}{\sqrt{a^2 + b^2 + c^2}}⁠, n=±ca2+b2+c2n = \pm\frac{c}{\sqrt{a^2 + b^2 + c^2}}⁠, with one sign for all three (Section 11.2). Lesson: direction cosines and direction ratios

2Squares of negative numbers

(−18)2=324(-18)^2 = 324 and (−4)2=16(-4)^2 = 16⁠: squaring removes the sign. Basics: Pythagoras twice

3The squares add up to 1

l2+m2+n2=1l^2 + m^2 + n^2 = 1 for every set of direction cosines. Basics: direction cosines

Solution

  1. Find a2+b2+c2\sqrt{a^2 + b^2 + c^2}

    a2+b2+c2=(−18)2+122+(−4)2=324+144+16=484=22.\begin{aligned} &\sqrt{a^2 + b^2 + c^2} \\ &= \sqrt{(-18)^2 + 12^2 + (-4)^2} \\ &= \sqrt{324 + 144 + 16} \\ &= \sqrt{484} = 22. \end{aligned}
    This number is the length of the vector −18i^+12j^−4k^-18\hat i + 12\hat j - 4\hat k⁠, which points along the line. Dividing by it will shrink that vector to length 1. Here 484=222484 = 22^2⁠, so the root is a whole number.
    Length of ai^+bj^+ck^a\hat i + b\hat j + c\hat k is a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠. Basics: Pythagoras twice

  2. Divide each direction ratio by 22

    l=−1822=−911,m=1222=611,n=−422=−211.\begin{aligned} l &= \frac{-18}{22} = -\frac{9}{11}, \\ m &= \frac{12}{22} = \frac{6}{11}, \\ n &= \frac{-4}{22} = -\frac{2}{11}. \end{aligned}
    The book writes l=akl = ak⁠, m=bkm = bk⁠, n=ckn = ck and finds k=±1a2+b2+c2k = \pm\frac{1}{\sqrt{a^2 + b^2 + c^2}} from l2+m2+n2=1l^2 + m^2 + n^2 = 1⁠. Taking k=122k = \frac{1}{22} keeps the signs of the given ratios. Each fraction is then cancelled by 2.
    l=aa2+b2+c2l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, and so on (Section 11.2). Lesson: direction cosines

  3. The other sign

    Taking k=−122k = -\frac{1}{22} instead gives

    911,−611,211.\frac{9}{11}, \quad -\frac{6}{11}, \quad \frac{2}{11}.
    Direction ratios describe a line, not a direction along it, so both signs of kk are possible: the two answers are the same line taken in opposite directions, and all three signs change together. The book's convention, as in its own examples, is to keep the signs of the given ratios, which is the answer in Step 2.
    The sign of kk is a choice, and it is the same for ll⁠, mm and nn (Section 11.2). Exercise 11.1 player

  4. Check, and a shortcut

    81121+36121+4121=121121=1.\frac{81}{121} + \frac{36}{121} + \frac{4}{121} = \frac{121}{121} = 1.

    Shortcut: −18,12,−4=2(−9,6,−2)-18, 12, -4 = 2(-9, 6, -2)⁠, so −9,6,−2-9, 6, -2 are also direction ratios, and 81+36+4=121=11\sqrt{81 + 36 + 4} = \sqrt{121} = 11 gives −911,611,−211-\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} at once.

    The squares adding to 1 confirms that these are direction cosines. The shortcut works because any nonzero multiple of a set of direction ratios is again a set of direction ratios (Section 11.2); taking out the common factor 2 first makes the numbers smaller.
    Proportional direction ratios describe the same line. Basics: direction cosines
Result

The direction cosines are −911,611,−211-\frac{9}{11}, \frac{6}{11}, -\frac{2}{11} (or 911,−611,211\frac{9}{11}, -\frac{6}{11}, \frac{2}{11} for the opposite direction).

(Textbook answer: −911,611,−211\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}⁠.)

Check it with numbers

l≈−0.818182l \approx -0.818182⁠, m≈0.545455m \approx 0.545455⁠, n≈−0.181818n \approx -0.181818⁠. Each times 22 gives back −18-18⁠, 1212⁠, −4-4⁠.

The direction angles are about 144.90∘144.90^\circ⁠, 56.94∘56.94^\circ and 100.48∘100.48^\circ⁠: obtuse with the xx and zz-axes, where the ratios are negative.

Common mistakes
  1. Dividing by the sum −18+12−4-18 + 12 - 4 instead of by a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠.
  2. Getting (−18)2(-18)^2 or (−4)2(-4)^2 negative. Squares are never negative: 324324 and 1616⁠.
  3. Changing only some signs for the other direction. Either keep all three signs or change all three.
What to take away
  1. Direction cosines = direction ratios divided by a2+b2+c2\sqrt{a^2 + b^2 + c^2}⁠.
  2. Take out any common factor of the ratios first; the answer does not change.
  3. The other sign gives the same line in the opposite direction.

Every fact used, and where it comes from

FactUsed inSource
Length a2+b2+c2\sqrt{a^2 + b^2 + c^2} of ai^+bj^+ck^a\hat i + b\hat j + c\hat kStep 1Section 10.5.1 (Basics)
l=±aa2+b2+c2l = \pm\frac{a}{\sqrt{a^2 + b^2 + c^2}}⁠, and so on, one sign for all threeSteps 2, 3Section 11.2 (Lesson)
Any two sets of direction ratios of a line are proportionalStep 4Section 11.2
l2+m2+n2=1l^2 + m^2 + n^2 = 1Step 4Section 11.2 (Basics)
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