13.3 Vector Operations and Magnitude

2026 📋 Syllabus Objectives

By the end of this topic, you should be able to:

  • Find the magnitude (size/length) of a vector
  • Add and subtract vectors
  • Multiply a vector by a scalar (a scalar is just a number)
  • Equate like vectors — compare the matching parts of two vectors to form equations and solve for unknowns
  • Solve problems using vector geometry — including finding unknown points, ratios on line segments, and proving that points are collinear (lie on the same straight line), using diagrams where needed

📖 Main Content

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Section 1: What is a Vector?

A vector is a quantity that has both a size (called magnitude) and a direction. We write vectors in two main ways:

Column vector form: v⃗=(34)\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} This means: move 3 units in the x-direction (horizontal) and 4 units in the y-direction (vertical).

i and j notation: v⃗=3i+4j\vec{v} = 3\mathbf{i} + 4\mathbf{j} Here, i means "1 unit in the x-direction" and j means "1 unit in the y-direction". They are called unit vectors along the axes.

Both forms mean exactly the same thing — just different ways of writing it.

💡 Think of it like directions on a map: "Go 3 steps East and 4 steps North." That's a vector!

Section 2: Magnitude of a Vector

The magnitude of a vector is its length — how far it actually travels, regardless of direction.

If a vector is v⃗=(xy)\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}, then its magnitude is written as ∣v⃗∣|\vec{v}| and calculated using the Pythagorean theorem:

∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2}

💡 Why Pythagoras? Because the x and y components form a right-angled triangle, and the vector itself is the hypotenuse (the longest side).

Example 1: Find the magnitude of v⃗=(34)\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}

∣v⃗∣=32+42=9+16=25=5|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

So the magnitude is 5 units.

Example 2: Find the magnitude of w⃗=−5i+12j\vec{w} = -5\mathbf{i} + 12\mathbf{j}

∣w⃗∣=(−5)2+122=25+144=169=13|\vec{w}| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

So the magnitude is 13 units.

⚠️ Important: Always square the number before adding — squaring a negative gives a positive, so (−5)2=25(-5)^2 = 25, not −25-25.

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