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Grade 12 · MCV4U · Calculus & Vectors

MCV4U Lines and Planes Practice Test (with Solutions)

Lines and Planes is the last MCV4U unit, and on Martian Lab it includes intersections, which some courses teach as a unit of their own. The free test is 8 multiple-choice questions on direction and normal vectors, switching between vector, parametric and symmetric equations, the plane through a line and a point, and how two planes meet. Pick your answer before you open it.

Free Lines and Planes Practice Test

8 questions, exactly as a free account gets them in the first Lines and Planes test on Martian Lab. Try each one on paper first, then open the answer.

  1. Which vector equation represents the line perpendicular to the line shown?
    Figure for the question
    1. r⃗=(0,−1)+s ⁣(1,2),s∈R\vec{r} = \left(0,-1\right)+s\!\left(1,2\right), s \in \mathbf{R}
    2. r⃗=(0,−1)+s ⁣(−2,1),s∈R\vec{r} = \left(0,-1\right)+s\!\left(-2,1\right), s \in \mathbf{R}
    3. r⃗=(0,−1)+s ⁣(2,1),s∈R\vec{r} = \left(0,-1\right)+s\!\left(2,1\right), s \in \mathbf{R}
    4. none of these
    Show answer
    b) r⃗=(0,−1)+s ⁣(−2,1),s∈R\vec{r} = \left(0,-1\right)+s\!\left(-2,1\right), s \in \mathbf{R}
  2. Which of the following is a direction vector for the line with scalar equation 3x+2y−1=03x +2y - 1 = 0?
    1. m⃗=[3,2]\vec{m} = [3,2]
    2. m⃗=[−2,3]\vec{m} = [-2,3]
    3. m⃗=[3,−2]\vec{m} = [3,-2]
    4. none of these
    Show answer
    b) m⃗=[−2,3]\vec{m} = [-2,3]
  3. Which of the following vectors is perpendicular to the line passing through the points P ⁣(2,3)P\!\left(2,3\right) and Q ⁣(7,10)Q\!\left(7,10\right)?
    1. v⃗=[5,7]\vec{v} = [5,7]
    2. v⃗=[7,−5]\vec{v} = [7,-5]
    3. v⃗=[−3,2]\vec{v} = [-3,2]
    4. v⃗=[−10,7]\vec{v} = [-10,7]
    Show answer
    b) v⃗=[7,−5]\vec{v} = [7,-5]
  4. Which of the following is the parametric equation of the line with symmetric equation x+33=y+22=z−12 ?\dfrac{x+3}{3} = \dfrac{y+2}{2} = \dfrac{z-1}{2}\,?
    1. x=3t−3, y=2t−2, z=2t+1, t∈Rx = 3t-3,\ y = 2t-2,\ z = 2t+1,\ t \in \mathbf{R}
    2. x=3t+3, y=2t+2, z=2t−1, t∈Rx = 3t+3,\ y = 2t+2,\ z = 2t-1,\ t \in \mathbf{R}
    3. x=3t−3, y=2t−2, z=t+2, t∈Rx = 3t-3,\ y = 2t-2,\ z = t+2,\ t \in \mathbf{R}
    4. none of these
    Show answer
    a) x=3t−3, y=2t−2, z=2t+1, t∈Rx = 3t-3,\ y = 2t-2,\ z = 2t+1,\ t \in \mathbf{R}
  5. Which of the following is not an equation for the line passing through the points P ⁣(1,4,−3)P\!\left(1,4,-3\right) and Q ⁣(3,2,1)Q\!\left(3,2,1\right)?
    1. r⃗=(1,4,−3)+s ⁣(2,−2,4),s∈R\vec{r} = \left(1,4,-3\right)+s\!\left(2,-2,4\right), s \in \mathbf{R}
    2. x=−t+3, y=t+2, z=−2t+1, t∈Rx = -t+3,\ y = t+2,\ z = -2t+1,\ t \in \mathbf{R}
    3. x−32=y−2−2=z+14\dfrac{x-3}{2} = \dfrac{y-2}{-2} = \dfrac{z+1}{4}
    4. r⃗=(3,2,1)+s ⁣(1,−1,2),s∈R\vec{r} = \left(3,2,1\right)+s\!\left(1,-1,2\right), s \in \mathbf{R}
    Show answer
    c) x−32=y−2−2=z+14\dfrac{x-3}{2} = \dfrac{y-2}{-2} = \dfrac{z+1}{4}
  6. A plane is defined by the equation 3x−2z=4y+13x - 2z = 4y + 1. Which of the following is the normal vector of this plane?
    1. n⃗=[3,−2,4]\vec{n} = [3,-2,4]
    2. n⃗=[3,4,−2]\vec{n} = [3,4,-2]
    3. n⃗=[3,2,4]\vec{n} = [3,2,4]
    4. n⃗=[3,−4,−2]\vec{n} = [3,-4,-2]
    Show answer
    d) n⃗=[3,−4,−2]\vec{n} = [3,-4,-2]
  7. Which of the following is the vector equation of a plane that contains the line x−35=y+2−2=z−13\dfrac{x-3}{5} = \dfrac{y+2}{-2} = \dfrac{z-1}{3} and the point P ⁣(1,2,3)P\!\left(1,2,3\right)?
    1. r⃗=(3,−2,1)+s ⁣(5,−2,3)+t ⁣(2,0,4), s,t∈R\vec{r} = \left(3,-2,1\right)+s\!\left(5,-2,3\right)+t\!\left(2,0,4\right),\ s,t \in \mathbf{R}
    2. r⃗=(3,−2,1)+s ⁣(5,−2,3)+t ⁣(−2,4,2), s,t∈R\vec{r} = \left(3,-2,1\right)+s\!\left(5,-2,3\right)+t\!\left(-2,4,2\right),\ s,t \in \mathbf{R}
    3. r⃗=(3,−2,1)+s ⁣(−2,4,2)+t ⁣(5,−2,3), s,t∈R\vec{r} = \left(3,-2,1\right)+s\!\left(-2,4,2\right)+t\!\left(5,-2,3\right),\ s,t \in \mathbf{R}
    4. r⃗=(3,−2,1)+s ⁣(1,2,3)+t ⁣(5,−2,3), s,t∈R\vec{r} = \left(3,-2,1\right)+s\!\left(1,2,3\right)+t\!\left(5,-2,3\right),\ s,t \in \mathbf{R}
    Show answer
    b) r⃗=(3,−2,1)+s ⁣(5,−2,3)+t ⁣(−2,4,2), s,t∈R\vec{r} = \left(3,-2,1\right)+s\!\left(5,-2,3\right)+t\!\left(-2,4,2\right),\ s,t \in \mathbf{R}
  8. The two planes 3x−2y−4z+1=03x - 2y - 4z + 1 = 0 and 2x−5y−3=02x - 5y - 3 = 0 are:
    1. perpendicular
    2. coincident
    3. parallel and distinct
    4. none of these
    Show answer
    d) none of these

This Unit on Martian Lab

The lessons go from equations of lines in 2D and 3D to equations of planes, then intersections of lines, of lines and planes, and of two and three planes.

  1. Equations of Lines in 2D
  2. Equations of Lines in 3D
  3. Equations of Planes
  4. Intersections of Lines
  5. Intersections of Lines & Planes
  6. Intersections of Two Planes
  7. Intersections of Three Planes

Then 4 full Lines and Planes practice tests to check yourself.

Start Lines and Planes free

Lines and Planes FAQ

What is on an MCV4U lines and planes test?+

Vector, parametric and symmetric equations of lines in 2D and 3D, vector and scalar equations of planes, normal vectors, and the intersections of lines and planes. The free questions on this page are on the equations and on how two planes relate; solving for intersection points is in the lessons.

Where are intersections of lines and planes?+

In this unit. Martian Lab teaches intersections of lines, of a line and a plane, of two planes and of three planes as the last four lessons of Lines and Planes, so there is no separate intersections unit.

How do you find the normal vector of a plane?+

Move every term to one side so the plane reads Ax + By + Cz + D = 0. The normal is [A, B, C]. For 3x - 2z = 4y + 1 that gives 3x - 4y - 2z - 1 = 0, so n = [3, -4, -2].

How do you turn a symmetric equation into parametric equations?+

Set each fraction equal to t and solve for x, y and z. (x + 3)/3 = (y + 2)/2 = (z - 1)/2 becomes x = 3t - 3, y = 2t - 2, z = 2t + 1.

How can you tell if two planes are parallel or perpendicular?+

Compare their normals. Parallel planes have normals that are scalar multiples of each other, and perpendicular planes have normals with a dot product of 0. If neither holds, the planes meet at some other angle along a line.

Lines and Planes Topics

One idea at a time: a short explanation, then five worked examples.

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