Multivariable Calculus Quiz: 3d Line Equations
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3d Line EquationsQuestion 1 of 1

Consider the line LL given by the symmetric equations x−32=y+1−1=z−54\frac{x-3}{2} = \frac{y+1}{-1} = \frac{z-5}{4}. Which of the following lines is parallel to LL and passes through the origin?

r⃗=t⟨2,−1,4⟩\vec{r} = t\langle 2, -1, 4 \rangle
x3=y−1=z5\frac{x}{3} = \frac{y}{-1} = \frac{z}{5}
r⃗=t⟨1,−2,2⟩\vec{r} = t\langle 1, -2, 2 \rangle
x1=y+1−2=z2\frac{x}{1} = \frac{y+1}{-2} = \frac{z}{2}
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Multivariable Calculus Quiz

Multivariable Calculus Quiz: 3d Line Equations

Practice 3d Line Equations in Multivariable Calculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on 3d Line Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Multivariable Calculus.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Consider the line LL given by the symmetric equations x−32=y+1−1=z−54\frac{x-3}{2} = \frac{y+1}{-1} = \frac{z-5}{4}. Which of the following lines is parallel to LL and passes through the origin?

  1. r⃗=t⟨2,−1,4⟩\vec{r} = t\langle 2, -1, 4 \rangle (correct answer)
  2. x3=y−1=z5\frac{x}{3} = \frac{y}{-1} = \frac{z}{5}
  3. r⃗=t⟨1,−2,2⟩\vec{r} = t\langle 1, -2, 2 \rangle
  4. x1=y+1−2=z2\frac{x}{1} = \frac{y+1}{-2} = \frac{z}{2}
Explanation: For lines to be parallel, they must have direction vectors that are scalar multiples of each other. The given line L has direction vector ⟨2, -1, 4⟩ from the denominators in the symmetric equation. A line through the origin has the form ⟨0, 0, 0⟩ + t⟨a, b, c⟩ = t⟨a, b, c⟩. Choice A has direction vector ⟨2, -1, 4⟩, which is exactly the same as L's direction vector, so it's parallel and passes through the origin. Choice B has direction vector ⟨3, -1, 5⟩, which is not a scalar multiple of ⟨2, -1, 4⟩. Choice C has direction vector ⟨1, -2, 2⟩, which is also not a scalar multiple of ⟨2, -1, 4⟩. Choice D doesn't pass through the origin since when x = z = 0, we get y = -1, not y = 0.