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Chapter 11 · Three Dimensional Geometry

Exercise 11.2 · Q6: Cartesian equation of the line through (−2,4,−5)(-2, 4, -5) parallel to x+33=y−45=z+86\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}

Exercise 11.2Cartesian equation of a line · Section 11.3.1
The question

Find the Cartesian equation of the line which passes through the point (−2,4,−5)(-2, 4, -5) and parallel to the line given by x+33=y−45=z+86\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6}⁠.

Exam answer

The given line x+33=y−45=z+86\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} has direction ratios 3,5,63, 5, 6⁠.

The required line is parallel to it, so it has the same direction ratios 3,5,63, 5, 6⁠.

The line through (x1,y1,z1)(x_1, y_1, z_1) with direction ratios a,b,ca, b, c is

x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}

Here (x1,y1,z1)=(−2,4,−5)(x_1, y_1, z_1) = (-2, 4, -5)⁠, so

x−(−2)3=y−45=z−(−5)6\frac{x - (-2)}{3} = \frac{y - 4}{5} = \frac{z - (-5)}{6}

∴\therefore The required line is x+23=y−45=z+56\dfrac{x + 2}{3} = \dfrac{y - 4}{5} = \dfrac{z + 5}{6}⁠.

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

The one idea this problem is built on

In the Cartesian form x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}⁠, the numerators carry the point and the denominators carry the direction. A parallel line keeps the direction, so it keeps the denominators; only the point in the numerators changes.

Given, to find, and the plan
Given
The point A(−2,4,−5)A(-2, 4, -5)⁠, and the line x+33=y−45=z+86\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} that the required line is parallel to.
To find
The Cartesian equation of the line through AA parallel to the given line.
Plan
(1) Read the direction ratios of the given line. (2) A parallel line has the same direction ratios. (3) Write the Cartesian equation through AA⁠. (4) Check, and see that it is a different line.

Basics you need

1Cartesian form of a line

The line through (x1,y1,z1)(x_1, y_1, z_1) with direction ratios a,b,ca, b, c is x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} (Section 11.3.1, equation (3)). Basics: a line is a point plus a direction

2Parallel lines

Parallel lines have proportional direction ratios, so the same set of direction ratios serves for both (Section 11.4, condition (ii)). Lesson: the angle between two lines

3Signs in the numerators

x+3=x−(−3)x + 3 = x - (-3)⁠: a plus sign in a numerator means a negative coordinate of the point. Basics: three numbers fix a point

Solution

  1. Direction ratios of the given line

    Write the given line in the standard form:

    x−(−3)3=y−45=z−(−8)6.\frac{x - (-3)}{3} = \frac{y - 4}{5} = \frac{z - (-8)}{6}.

    It passes through (−3,4,−8)(-3, 4, -8) and has direction ratios 3,5,63, 5, 6⁠.

    Matching with x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}⁠, the denominators are the direction ratios. The point (−3,4,−8)(-3, 4, -8) is not needed for this question, but reading it correctly (x+3x + 3 means x1=−3x_1 = -3⁠) helps in Step 4.
    Section 11.3.1, equation (3). Lesson: the equation of a line

  2. A parallel line has the same direction ratios

    The required line is parallel to the given line, so 3,5,63, 5, 6 are also its direction ratios.

    Parallel lines have proportional direction ratios: a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}⁠. Taking the factor 1, the same numbers 3,5,63, 5, 6 can be used for the new line. Any nonzero multiple, such as 6,10,126, 10, 12⁠, would also be correct but less simple.
    Section 11.4, condition (ii). Lesson: perpendicular and parallel lines

  3. Write the equation through (−2,4,−5)(-2, 4, -5)

    x−(−2)3=y−45=z−(−5)6,that isx+23=y−45=z+56.\begin{aligned} &\frac{x - (-2)}{3} = \frac{y - 4}{5} = \frac{z - (-5)}{6}, \\ &\text{that is}\quad \frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}. \end{aligned}
    Put (x1,y1,z1)=(−2,4,−5)(x_1, y_1, z_1) = (-2, 4, -5) and (a,b,c)=(3,5,6)(a, b, c) = (3, 5, 6) into the Cartesian form. Subtracting a negative gives a plus: x−(−2)=x+2x - (-2) = x + 2 and z−(−5)=z+5z - (-5) = z + 5⁠.
    Cartesian equation of a line. Exercise 11.2 player

  4. Check, and a different line

    At (−2,4,−5)(-2, 4, -5) all three fractions are 03=05=06=0\frac{0}{3} = \frac{0}{5} = \frac{0}{6} = 0⁠, so the point is on the line. In vector form the line is r⃗=−2i^+4j^−5k^+λ(3i^+5j^+6k^)\vec r = -2\hat i + 4\hat j - 5\hat k + \lambda(3\hat i + 5\hat j + 6\hat k)⁠.

    The point (−3,4,−8)(-3, 4, -8) of the given line gives −3+23=−13\frac{-3 + 2}{3} = -\frac13 but 4−45=0\frac{4 - 4}{5} = 0⁠: not equal, so it is not on the new line.

    (−3, 4, −8) A(−2, 4, −5) given line required line
    Oblique projection, not to scale. The given line through (−3,4,−8)(-3, 4, -8) and the required line through A(−2,4,−5)A(-2, 4, -5) both have direction ratios 3,5,63, 5, 6⁠: two different parallel lines.
    A point is on the line exactly when the three fractions are equal (they all equal the same λ\lambda⁠). The given point makes all three 0. The two lines have the same direction, and a point of one is not on the other, so they are distinct parallel lines, as the question intends.
    Each point of the line has one value of λ\lambda⁠. Basics: go to A, then walk λ copies of b
Result

The required line is x+23=y−45=z+56\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}⁠.

(Textbook answer: x+23=y−45=z+56\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}⁠.)

Check it with numbers

Take λ=0.4\lambda = 0.4 on the new line: the point is (−2+1.2,4+2,−5+2.4)(-2 + 1.2, 4 + 2, -5 + 2.4) =(−0.8,6,−2.6)= (-0.8, 6, -2.6)⁠. Then −0.8+23=0.4\frac{-0.8 + 2}{3} = 0.4⁠, 6−45=0.4\frac{6 - 4}{5} = 0.4 and −2.6+56=0.4\frac{-2.6 + 5}{6} = 0.4⁠.

The direction 3,5,63, 5, 6 has length 9+25+36=70≈8.366600\sqrt{9 + 25 + 36} = \sqrt{70} \approx 8.366600 for both lines.

Common mistakes
  1. Taking the numbers in the numerators as direction ratios. The direction ratios are the denominators 3,5,63, 5, 6⁠.
  2. Sign slips with the point: x−(−2)=x+2x - (-2) = x + 2⁠, not x−2x - 2⁠; z−(−5)=z+5z - (-5) = z + 5⁠.
  3. Copying the point of the given line into the answer. The new line passes through (−2,4,−5)(-2, 4, -5)⁠, not (−3,4,−8)(-3, 4, -8)⁠.
What to take away
  1. In x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}⁠, the point is in the numerators and the direction in the denominators.
  2. Parallel line: keep the denominators, change the point.
  3. A plus sign in a numerator means a negative coordinate.

Every fact used, and where it comes from

FactUsed inSource
Cartesian equation x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}⁠; its denominators are direction ratiosSteps 1, 3Section 11.3.1, equation (3) (Lesson)
Parallel lines: a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}Step 2Section 11.4, condition (ii) (Lesson)
Vector form r⃗=a⃗+λb⃗\vec r = \vec a + \lambda\vec bStep 4Section 11.3.1, equation (1) (Basics)
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