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Learning Resources

Vectors (Basic)

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Form 3 Pathway: N/A

First Principles

Objective

Perform basic vector addition, subtraction, scalar multiplication, and calculate vector magnitude.

Concrete Scenario: A medical drone leaves a depot on the shore of Lake Victoria. The pilot commands it to fly 3 units East and 1 unit North \(\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\). A crosswind pushes the drone 1 unit East and 2 units South \(\mathbf{b} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\). Where does the drone actually land? To find out, we add the two displacement vectors.

Geometric & Algebraic Insight:

  • Tip-to-Tail Rule: Draw vector \(\mathbf{a}\). From the tip (end) of \(\mathbf{a}\), draw vector \(\mathbf{b}\). The vector from the start of \(\mathbf{a}\) to the tip of \(\mathbf{b}\) is the resultant vector \(\mathbf{a} + \mathbf{b}\).
  • Column Vectors: Adding vectors means adding horizontal steps together and vertical steps together: \(\begin{pmatrix} a_x \\ a_y \end{pmatrix} + \begin{pmatrix} b_x \\ b_y \end{pmatrix} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix}\).
  • Vector Magnitude: The length of vector \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}\) is found using Pythagoras: \(|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}\).

Vector Addition (Tip-to-Tail Method)

a = (3, 1)b = (1, -2)Resultant = (4, -1)Start

Key Formulas

\[\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix} = v_x\mathbf{i} + v_y\mathbf{j}\] — Column & Unit Vector Form: Expressing horizontal and vertical components.
\[|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}\] — Vector Magnitude: Absolute length calculated via Pythagoras.
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix}, \quad \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_x - b_x \\ a_y - b_y \end{pmatrix}\] — Vector Addition & Subtraction.
\[k\mathbf{v} = \begin{pmatrix} k v_x \\ k v_y \end{pmatrix}\] — Scalar Multiplication: Scaling magnitude by factor \(k\) while maintaining or reversing direction.

Worked Examples

Example 1 (Easy) — Vector Addition & Magnitude:

Problem: Given \(\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\), find \(\mathbf{a} + \mathbf{b}\) and its magnitude.

  1. Add components: \[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 3 + 1 \\ 1 + 4 \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\]
  2. Calculate magnitude: \[|\mathbf{a} + \mathbf{b}| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40\]
Example 2 (Medium) — Scalar Multiplication & Subtraction:

Problem: Given \(\mathbf{u} = 2\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{v} = -\mathbf{i} + 3\mathbf{j}\), calculate \(2\mathbf{u} - \mathbf{v}\).

  1. Scale vector \(\mathbf{u}\): \[2\mathbf{u} = 2(2\mathbf{i} + 5\mathbf{j}) = 4\mathbf{i} + 10\mathbf{j}\]
  2. Subtract \(\mathbf{v}\): \[2\mathbf{u} - \mathbf{v} = (4 - (-1))\mathbf{i} + (10 - 3)\mathbf{j} = 5\mathbf{i} + 7\mathbf{j}\]
Example 3 (Hard) — Bearing & Vector Resultant:

Problem: Vector \(\mathbf{A} = 3\mathbf{i} - 2\mathbf{j}\). Vector \(\mathbf{B}\) has magnitude 5 at an angle of 60° to the positive x-axis. Find the x-component of \(\mathbf{A} + \mathbf{B}\).

  1. Find x-component of vector \(\mathbf{B}\): \[B_x = 5 \cos(60^{\circ}) = 5(0.5) = 2.5\]
  2. Add x-components: \(A_x + B_x = 3 + 2.5 = 5.5\).

Common Mistakes

Mistake Adding magnitudes directly instead of adding vector components (e.g., assuming \(|(3,0)| + |(0,4)| = 3 + 4 = 7\)).
Correction Vectors must be added component-wise first: \((3,0) + (0,4) = (3,4)\), then compute total magnitude \(\sqrt{3^2 + 4^2} = 5\).
Why it feels right Treating vector lengths like scalar values instead of taking directional orientation into account.
Mistake Thinking scalar multiplication changes a vector's direction angle.
Correction Multiplying by a positive scalar \(k\) only changes the length. It keeps the direction identical.
Why it feels right Confusing scalar multiplication with vector rotation.

Real World

Aviation & Drones: A drone flying over Lake Victoria must add its engine velocity vector to the wind velocity vector to determine true ground track and speed.
Maritime Navigation: Ferry captains crossing the Likoni Channel compute river current vectors to adjust their steering angle and arrive directly at the dock.
Transport & Logistics: Calculating total net displacement vectors for matatus navigating multi-stop delivery routes in urban traffic.

Practice

What is the magnitude of the displacement vector \( (3, 4) \)? (Type only the number, e.g., 5)
Review the concepts above.
Vector \(\mathbf{A}\) has components \(3\mathbf{i} - 4\mathbf{j}\) and vector \(\mathbf{B}\) has components \(-2\mathbf{i} + 5\mathbf{j}\). What is the x-component (i-component) of the resultant vector \(\mathbf{A} + \mathbf{B}\)? (Type only the number, e.g., 1)
Review the concepts above.
A matatu's displacement is represented by vector \(\mathbf{A} = 6\mathbf{i} - 8\mathbf{j}\) (in km). What is the magnitude of vector \(\mathbf{A}\)? (Type only the number, e.g., 10)
Review the concepts above.
Vector \(\mathbf{A}\) has components \( (3, -2) \). Vector \(\mathbf{B}\) has a magnitude of 5 units and makes an angle of 60\u00B0 with the positive x-axis. Find the x-component of the resultant vector \(\mathbf{A} + \mathbf{B}\). (Type only the number, e.g., 5.5)
Review the concepts above.
Given vector \(\mathbf{u} = (4, -2)\) and vector \(\mathbf{v} = (1, 5)\), find the y-component of vector \(2\mathbf{u} + \mathbf{v}\). (Type only the number, e.g., 1)
Review the concepts above.
A boat travels \(5\,\text{km}\) East and \(12\,\text{km}\) North. What is the total magnitude of its displacement in km? (Type only the number, e.g., 13)
Review the concepts above.