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Coordinate geometry and vectors

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 9 questions

Name: ____________________   Date: ____________

  1. What is the distance between the points (1, 2) and (4, 6)?

    • A 5
    • B 3
    • C 4
    • D 7

  2. What is the slope of the line through (1, 2) and (3, 8)?

    • A 3
    • B 2
    • C 4
    • D 6

  3. Which is the equation of a circle with centre at the origin and radius 5?

    • A x² + y² = 5
    • B x² - y² = 25
    • C x + y = 25
    • D x² + y² = 25

  4. What is the midpoint of the points (2, 4) and (6, 10)?

    • A (3, 5)
    • B (4, 6)
    • C (8, 14)
    • D (4, 7)

  5. What is the dot product of the vectors (1, 2, 3) and (4, -5, 6)?

    • A 32
    • B 0
    • C -12
    • D 12

  6. What is the magnitude of the vector (3, 4)?

    • A 7
    • B 1
    • C 5
    • D 25

  7. What is i x j, the cross product of the unit vectors i and j?

    • A -k
    • B k
    • C 0
    • D 1

  8. What is the focus of the parabola y² = 8x?

    • A (2, 0)
    • B (0, 2)
    • C (8, 0)
    • D (4, 0)

  9. Two vectors are perpendicular when their dot product is:

    • A Equal to their magnitudes
    • B 0
    • C -1
    • D 1

Answer key

1. A, 2. A, 3. D, 4. D, 5. D, 6. C, 7. B, 8. A, 9. B

  1. Distance = sqrt(3² + 4²) = sqrt(25) = 5.

  2. Slope = (8 - 2) / (3 - 1) = 6 / 2 = 3.

  3. A circle with centre (0, 0) and radius r has the equation x² + y² = r².

  4. The midpoint is ((2 + 6) / 2, (4 + 10) / 2) = (4, 7).

  5. 1 x 4 + 2 x (-5) + 3 x 6 = 4 - 10 + 18 = 12.

  6. The magnitude is sqrt(9 + 16) = 5.

  7. By the right-hand rule, i x j = k.

  8. Comparing with y² = 4ax gives 4a = 8, so a = 2 and the focus is (2, 0).

  9. a . b = |a||b| cos(theta), which is 0 exactly when the angle is 90 degrees.