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Learning Resources

Vectors I

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

Grade 10 Pathway: N/A

First Principles

Objective: Represent 2D vectors algebraically using column vectors and component unit vectors (\(\mathbf{i}, \mathbf{j}\)), and solve vector addition and subtraction geometrically using the tip-to-tail method.

Interactive Tip-to-Tail Vector Visualizer

Adjust horizontal and vertical components of Vector A and Vector B to observe resultant displacement!

a b R = a + b

(a) Concrete Scenario: A boda-boda rider in Nairobi delivers a package by traveling \(6\text{ km}\) due east, then turning and riding \(8\text{ km}\) due north. Although the rider traveled a total road distance of \(14\text{ km}\), their straight-line displacement vector from the origin is the hypotenuse vector \(\mathbf{R}\), measuring exactly \(10\text{ km}\).

(b) Key Vector Concepts:

  • Vector vs. Scalar: A scalar has magnitude only (e.g., speed = \(50\text{ km/h}\)); a vector has both magnitude and direction (e.g., velocity = \(50\text{ km/h}\) East).
  • Tip-to-Tail Method: To add vectors geometrically, place the tail (start) of the second vector at the tip (arrowhead) of the first. The resultant vector \(\mathbf{R}\) points from the initial start to the final tip.
  • Column Vector Notation: \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the horizontal and vertical axes.

(c) Vector Algebra Rules: \[\text{Addition: } \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\] \[\text{Subtraction: } \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}\] \[\text{Magnitude: } |\mathbf{a}| = \sqrt{x^2 + y^2}\]

Important Rule: NEVER add vector magnitudes directly unless both vectors point in the exact same direction! Always sum component \(x\)'s and component \(y\)'s separately first.

Key Formulas

\(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} = x\mathbf{i} + y\mathbf{j}\) — Column and unit vector component representation.
\(|\mathbf{a}| = \sqrt{x^2 + y^2}\) — Magnitude (length) of a 2D vector using Pythagoras' theorem.
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_a + x_b \\ y_a + y_b \end{pmatrix}\) — Addition of component vectors.
\(\mathbf{a} - \mathbf{b} = \begin{pmatrix} x_a - x_b \\ y_a - y_b \end{pmatrix}\) — Subtraction of component vectors.
\(\theta = \tan^{-1}\left(\frac{y}{x}\right)\) — Direction angle relative to the positive horizontal axis.

Worked Examples

Example 1 (Easy - Resultant Displacement): A matatu travels \(60\text{ km}\) due east from Nairobi, then turns and travels \(80\text{ km}\) due north. Calculate the magnitude of its total displacement in kilometres.
  1. Represent as column vectors: \(\mathbf{d}_1 = \begin{pmatrix} 60 \\ 0 \end{pmatrix}\), \(\mathbf{d}_2 = \begin{pmatrix} 0 \\ 80 \end{pmatrix}\).
  2. Sum component vectors: \(\mathbf{R} = \mathbf{d}_1 + \mathbf{d}_2 = \begin{pmatrix} 60 \\ 80 \end{pmatrix}\).
  3. Calculate magnitude: \(|\mathbf{R}| = \sqrt{60^2 + 80^2} = \sqrt{3,600 + 6,400} = \sqrt{10,000} = 100\text{ km}\).
  4. Answer: \(100\text{ km}\).
Example 2 (Medium - Vector Addition of Column Vectors): Nduku, a boda-boda rider, rides along vector \(\mathbf{u} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\) and then along vector \(\mathbf{v} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\). Calculate the magnitude of his total displacement vector \(\mathbf{R}\) to 2 decimal places.
  1. Add components: \(\mathbf{R} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 3+2 \\ 4+1 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}\).
  2. Calculate magnitude: \(|\mathbf{R}| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.071\).
  3. Answer: \(7.07\).
Example 3 (Hard - Vector Subtraction & Relative Position): In a drone flight system, position vector \(\mathbf{p} = 4\mathbf{i} + 7\mathbf{j}\) and position vector \(\mathbf{q} = 9\mathbf{i} - 5\mathbf{j}\). Find the displacement vector \(\vec{PQ} = \mathbf{q} - \mathbf{p}\) and calculate its magnitude.
  1. Subtract components: \(\vec{PQ} = (9 - 4)\mathbf{i} + (-5 - 7)\mathbf{j} = 5\mathbf{i} - 12\mathbf{j}\).
  2. Represent as column vector: \(\begin{pmatrix} 5 \\ -12 \end{pmatrix}\).
  3. Calculate magnitude: \(|\vec{PQ}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).
  4. Answer: Vector = \(5\mathbf{i} - 12\mathbf{j}\) ; Magnitude = \(13\).

Common Mistakes

Mistake Scalar addition of magnitudes: assuming \(|\mathbf{a} + \mathbf{b}| = |\mathbf{a}| + |\mathbf{b}|\).
Correction Magnitudes can only be added directly if vectors are strictly parallel pointing in the same direction! By the Triangle Inequality, \(|\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|\).
Why it feels right Traveling 6 km east then 8 km north covers 14 km total distance, tempting students to claim 14 km displacement instead of 10 km.
Mistake Subtracting vector components in reverse order when finding displacement between two points.
Correction Displacement from P to Q is ALWAYS \(\vec{PQ} = \mathbf{q} - \mathbf{p}\) (Final minus Initial).

Real World

Drone & Aviation Flight Paths: Navigation algorithms at Wilson Airport combine air speed vectors with wind velocity vectors using vector addition to determine true ground track speeds.
Marine Navigation: Cargo ships crossing the Mombasa port harbor correct for ocean tidal current vectors to maintain proper course direction.
Civil Structural Engineering: Engineers compute resultant load vectors on suspension bridge guy-wires in infrastructure projects across Kenya.

Practice

A matatu travels 60 km due east from Nairobi and then 80 km due north to a nearby town. What is the magnitude of its total displacement in kilometres? (Type only the number, e.g., 100)
Review the concepts above.
A delivery drone flies 9 km due west and then 12 km due south. What is the magnitude of its resultant displacement vector in kilometres? (Type only the number, e.g., 15)
Review the concepts above.
A boat crosses a river by traveling 3 km east while the current pushes it 4 km north. What is the magnitude of its total displacement in kilometres? (Type only the number, e.g., 5)
Review the concepts above.
Nduku, a boda-boda rider, rides along vector a = (3, 4) and then along vector b = (2, 1). What is the magnitude of his total displacement vector to two decimal places? (Type only the number, e.g., 7.07)
Review the concepts above.
Baraka constructs a fence with vector section a = (15, 0) metres and vector section b = (0, 10) metres. The resultant vector c = a + b. What is the length of section c in metres to two decimal places? (Type only the number, e.g., 18.03)
Review the concepts above.
Given vectors p = 4i + 7j and q = 9i - 5j, find the magnitude of the displacement vector PQ = q - p. (Type only the number, e.g., 13)
Review the concepts above.