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Vectors · Ages 17–18

Finding the Angle Between Vectors Practice

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Designed for 12th grade mathematics students. Calculate the dot products, magnitudes, and angles for each pair of vectors to the nearest tenth of a degree.

  1. Find the dot product of u = (3, 4) and v = (4, -3).

  2. Determine whether u = (3, 4) and v = (4, -3) are perpendicular.

  3. Find the magnitude of vector a = (2, -3, 6).

  4. Find the angle between u = (1, 0) and v = (1, 1).

  5. Find the angle between a = (2, 1, -2) and b = (1, 2, 2).

  6. Find the angle between u = (3, -1, 2) and v = (-1, 2, 5).

  7. Find the angle between p = (2, -4, 4) and q = (-1, 2, -2).

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Finding the Angle Between Vectors Practice

Answer key

  1. Find the dot product of u = (3, 4) and v = (4, -3).

    u · v = (3)(4) + (4)(-3) = 12 - 12 = 0.

  2. Determine whether u = (3, 4) and v = (4, -3) are perpendicular.

    Yes, because their dot product is 0, the angle is 90°.

  3. Find the magnitude of vector a = (2, -3, 6).

    |a| = √(2² + (-3)² + 6²) = √(4 + 9 + 36) = √49 = 7.

  4. Find the angle between u = (1, 0) and v = (1, 1).

    u · v = 1; |u| = 1; |v| = √2; cos(θ) = 1/√2; θ = 45.0°.

  5. Find the angle between a = (2, 1, -2) and b = (1, 2, 2).

    a · b = 2(1) + 1(2) + (-2)(2) = 0; cos(θ) = 0; θ = 90.0°.

  6. Find the angle between u = (3, -1, 2) and v = (-1, 2, 5).

    u · v = -3 - 2 + 10 = 5; |u| = √14; |v| = √30; cos(θ) = 5 / √(420) ≈ 0.2439; θ ≈ 75.9°.

  7. Find the angle between p = (2, -4, 4) and q = (-1, 2, -2).

    p · q = -2 - 8 - 8 = -18; |p| = 6; |q| = 3; cos(θ) = -18 / 18 = -1; θ = 180.0° (opposite directions).