Vectors in the Plane — Topic Review

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15 exam-style questions covering all lessons in this topic. Click Show Answer to reveal the full worked solution.

1. Fluency

   Q1 — Magnitude and Unit Vector

   Given v = 5i − 12j, find:   (a) |v|    (b) the unit vector v̂    (c) a vector of magnitude 3 in the direction of v.

2. Fluency

   Q2 — Vector Between Points and Midpoint

   Given P = (−2, 3) and Q = (4, 7), find:   (a) PQ    (b) |PQ|    (c) the midpoint M of PQ.

3. Understanding

   Q3 — Find Unknown Scalar

   Find all values of k such that |ki + 3j| = 5.

4. Fluency

   Q4 — Vector Arithmetic

   Given a = 3i + j and b = i − 2j, find:   (a) 2a − b    (b) |3a + 2b|

5. Understanding

   Q5 — Triangle Midsegment Theorem

   Triangle with A = (1, 0), B = (4, 4), C = (7, 3). M is the midpoint of AB and N is the midpoint of BC. Find MN and show it is parallel to AC with half its length.

6. Problem Solving

   Q6 — Parallelogram Vertices

   ABCD is a parallelogram with A = (2, 1), B = (5, 2), D = (3, 4). Find:   (a) the coordinates of C    (b) the intersection of the diagonals.

7. Fluency

   Q7 — Dot Product and Perpendicularity

   Given a = 2i + 3j and b = 6i − 4j, find a·b and determine whether a and b are perpendicular.

8. Fluency

   Q8 — Angle Between Vectors

   Find the angle between u = 3i + 4j and v = 5i, giving your answer to the nearest degree.

9. Understanding

   Q9 — Vector Projection

   Find the vector projection of a = 4i + 3j onto b = 2i.

10. Understanding

    Q10 — Centroid of a Triangle

    In triangle OAB, let OA = a and OB = b. M is the midpoint of AB. Show that the median OM passes through the point G = ⅓(a + b) and that the median from A also passes through G.

11. Understanding

    Q11 — Prove ABCD is a Parallelogram

    Points A = (1, 1), B = (5, 2), C = (4, 5), D = (0, 4). Use vectors to:   (a) prove ABCD is a parallelogram    (b) show the diagonals bisect each other.

12. Problem Solving

    Q12 — Angle in a Semicircle

    Let O be the centre of a circle of radius r. A and B are the endpoints of a diameter, so OA = a and OB = −a (with |a| = r). P is any point on the circle, so OP = p with |p| = r. Prove that ∠APB = 90°.

13. Fluency

    Q13 — Write Parametric Equations

    Write the parametric equations for the line through A = (3, −2) with direction d = i + 4j. State the coordinates of the point when t = 2.

14. Understanding

    Q14 — Find Intersection of Two Lines

    Find the intersection of ℓ₁: r = (1, 3) + t(2, −1) and ℓ₂: r = (5, 1) + s(1, 2).

15. Problem Solving

    Q15 — Line Through Two Points — Full Analysis

    A line passes through A = (1, 4) and B = (4, 2).   (a) Write its parametric equations.   (b) Find where it crosses the x-axis.   (c) Find the foot of the perpendicular from the origin O = (0, 0) to the line.