Practice Maths

Vectors in Three Dimensions — Topic Review

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This review covers all lessons in Vectors in Three Dimensions: 3D vector operations and magnitude, dot and cross products, vector equations of lines in 3D, and planes and intersections.

Review Questions

  1. Given a = (2, −1, 3) and b = (1, 4, −2): (a) Find |a| and |b|. (b) Find 3a − 2b. (c) Find the unit vector in the direction of a.

  2. Points A = (1, 2, −1), B = (4, 0, 3), C = (2, −1, 1). Find the perimeter of triangle ABC, giving your answer correct to 2 decimal places.

  3. Find the angle between the vectors u = (1, 2, 2) and v = (2, −1, 2), giving your answer in degrees correct to 1 decimal place.

  4. Find the value of k such that a = (k, 2, −1) and b = (3, k, 4) are perpendicular. Hence find the vector projection of a onto c = (1, 0, 1) when k takes this value.

  5. Let a = (1, 3, −2) and b = (2, 0, 1). (a) Find a × b. (b) Verify that a × b is perpendicular to both a and b. (c) Find the area of the parallelogram with sides a and b.

  6. Find the volume of the parallelepiped defined by vectors a = (2, 1, 0), b = (1, −1, 2), c = (0, 3, 1) using the scalar triple product.

  7. Write the vector, parametric, and Cartesian equations of the line through A = (2, −1, 3) and B = (5, 1, −1).

  8. Determine whether the lines L1: r = (1, 2, 3) + λ(2, −1, 1) and L2: r = (3, 1, 4) + μ(4, −2, 2) are the same line, parallel, intersecting, or skew. Justify your answer.

  9. Find the equation of the plane through points P = (1, 0, −1), Q = (2, 3, 0), R = (0, 1, 2). Hence verify that all three points satisfy the equation.

  10. Find the distance from the point Q = (3, −2, 5) to the plane 4x + 3y − 12z = 2. Give your answer as an exact fraction.

  11. Find the point of intersection of the line r = (0, 2, 1) + λ(1, −1, 3) and the plane 2x + y − z = 4.

  12. Find the line of intersection of the planes Π1: x + y + z = 6 and Π2: 2x − y + 3z = 10. Give the answer in vector form.

  13. Find the angle between the planes 3x − y + 2z = 5 and x + 2y − z = 1. Give your answer to the nearest degree.

  14. A plane Π has equation x − 2y + 2z = 9. A line L has equation r = (1, 1, 1) + λ(1, −2, 2). (a) Show that L is parallel to Π. (b) Find the distance from L to Π.

  15. The planes Π1: 2x + y + z = 4, Π2: x − y + 2z = 5, and Π3: 3x + 2y − z = k meet at a single point. Find k and the coordinates of that point.

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