Practice Maths

Vector Operations and Scalar Multiples

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Key Terms

Addition
a + b: add corresponding components; (a1, a2) + (b1, b2) = (a1+b1, a2+b2).
Scalar multiplication
ka: multiply each component by k; scales the magnitude by |k|; reverses direction if k < 0.
Subtraction
a − b = a + (−b); subtract corresponding components.
Zero vector 0
The vector (0, 0); a + 0 = a (additive identity).
Parallel vectors
a ∥ b if and only if b = ka for some non-zero scalar k.
Triangle law
Place the tail of b at the head of a; the resultant a + b goes from the tail of a to the head of b.

Vector Operations — Key Formulas

Let a = a1i + a2j and b = b1i + b2j, and let k be a scalar.

Addition:   a + b = (a1 + b1)i + (a2 + b2)j

Subtraction:   a − b = (a1 − b1)i + (a2 − b2)j

Scalar multiplication:   ka = ka1i + ka2j

Magnitude of scalar multiple:   |ka| = |k| × |a|

Vector between points:   AB = OB − OA

Midpoint:   OM = ½(OA + OB)

Parallel vectors: b = ka for some scalar k ≠ 0   (same or opposite direction, possibly different magnitude)

Collinear points: A, B, C are collinear if AC = k · AB for some scalar k

Worked Example 1 — Addition, Subtraction and Scalar Multiply

Given a = 3i + 2j and b = −i + 4j, find a + b, a − b, and 3a.

a + b = (3 − 1)i + (2 + 4)j = 2i + 6j

a − b = (3 − (−1))i + (2 − 4)j = 4i − 2j

3a = 3(3i + 2j) = 9i + 6j

Worked Example 2 — Midpoint

Find the midpoint M of AB where A = (2, 5) and B = (8, 3).

OA = 2i + 5j,   OB = 8i + 3j

OM = ½(OA + OB) = ½(10i + 8j) = 5i + 4j

So M = (5, 4).

Worked Example 3 — Collinearity

Show A = (0, 0), B = (2, 3), C = (6, 9) are collinear.

AB = OB − OA = 2i + 3j

AC = OC − OA = 6i + 9j = 3(2i + 3j) = 3AB

Since AC = 3AB, the vectors are parallel and share point A, so A, B, C are collinear. ✓

Hot Tip — Direction of Scalar Multiples: If k > 0, then ka points in the same direction as a. If k < 0, it points in the opposite direction. The magnitude scales by |k|, regardless of sign.

The Geometry of Vector Addition

Vector addition has a beautiful geometric interpretation. There are two equivalent geometric rules:

Triangle Rule (tip-to-tail): Place the tail of b at the head (tip) of a. The resultant vector a + b goes from the tail of a to the head of b. It “closes the triangle.”

Parallelogram Rule (tail-to-tail): Place both vectors with their tails at the same point. Complete the parallelogram. The diagonal from the common tail point is a + b.

Both rules give the same result. Algebraically, adding vectors just means adding corresponding components, which is easy to compute.

Vector Subtraction

a − b = a + (−b). Geometrically, reverse the direction of b (negate it), then add tip-to-tail. Alternatively, in the parallelogram, a − b is the OTHER diagonal — from the head of b to the head of a.

Important application: the vector from point A to point B is AB = OB − OA. You subtract the starting point’s position vector from the ending point’s position vector.

Scalar Multiplication

Multiplying a vector a by scalar k scales its magnitude by |k| and preserves or reverses its direction:

• k = 2: same direction, twice as long.

• k = ½: same direction, half as long.

• k = −3: opposite direction, three times as long.

• k = 0: result is the zero vector 0.

Parallel Vectors

Two non-zero vectors a and b are parallel if and only if b = ka for some scalar k. This means their directions are the same or exactly opposite.

In component form: a = a1i + a2j and b = b1i + b2j are parallel iff a1b2 = a2b1 (their components are proportional).

Collinear Points

Three points A, B, C are collinear (lie on a single straight line) iff AC = k · AB for some scalar k. This says that the vector from A to C is a scalar multiple of the vector from A to B — meaning they point in the same (or opposite) direction and share a common starting point.

Important: two vectors being parallel alone is not enough for collinearity — they must also share a common point.

Linear Combinations

A linear combination of vectors a and b is any vector of the form ma + nb. If a and b are not parallel, every vector in the plane can be written as a unique linear combination of a and b. To find the scalars m and n, equate components and solve the system of two equations.

Mastery Practice

  1. Fluency

    Q1 — Basic Vector Arithmetic

    Given a = 4i − j and b = −2i + 3j, find:   (a) a + b    (b) a − b    (c) 2a + 3b

  2. Fluency

    Q2 — Vector Between Two Points

    Find AB where A = (3, 7) and B = (1, 2).

  3. Fluency

    Q3 — Scalar Multiplication

    Compute:   (a) 3(2i + 5j)    (b) −2(i − 4j)    (c) 0.5(−6i + 10j)

  4. Fluency

    Q4 — Midpoint

    Find the midpoint of the segment from P(1, 4) to Q(7, 2).

  5. Understanding

    Q5 — Finding an Unknown Vector

    Given p = i + 2j and q = 3i − j, find r such that p + r = q.

  6. Understanding

    Q6 — Parallel Vectors

    Show that a = 6i − 4j and b = −9i + 6j are parallel. State the scalar.

  7. Understanding

    Q7 — Collinear Points

    Show that A = (1, 2), B = (4, 5), C = (10, 11) are collinear.

  8. Understanding

    Q8 — Linear Combination

    Express r = 7i + j as a linear combination of a = 2i + j and b = i − j.

  9. Problem Solving

    Q9 — Parallelogram Vertex

    ABCD is a parallelogram with A = (1, 1), B = (4, 3), C = (6, 7). Find the coordinates of D.

  10. Problem Solving

    Q10 — Midpoints and Parallel Vectors

    M is the midpoint of AB and N is the midpoint of BC. Given A = (0, 4), B = (6, 2), C = (8, 8), find MN as a vector and show it is parallel to AC with half its length.

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