Scalars & Vectors: Resultants and Components
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Lesson overview
Telling scalars from vectors, finding the resultant of two vectors and resolving a vector into perpendicular components.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
Some physical quantities are completely described by a number and a unit: 5 kg of rice, a temperature of 30 °C, a time of 20 s. Others are not. If a friend tells you to walk 200 m from the school gate, you still need to know which way to go. Physics separates these two kinds of quantity because they combine in different ways. Two masses of 3 kg and 4 kg always add up to 7 kg. Two forces of 3 N and 4 N, however, can give a resultant anywhere from 1 N to 7 N, depending on the angle between them. In this lesson you will learn to tell scalars from vectors, add two vectors using the triangle and parallelogram laws, and split a single vector into two perpendicular components. These skills return in projectiles, inclined planes and equilibrium, so UTME tests them every year.
Key concept: scalars, vectors and the resultant
A scalar quantity has magnitude (size) only. Examples: mass, distance, speed, time, energy, work, power, temperature, density, volume and electric charge. A vector quantity has both magnitude and direction. Examples: displacement, velocity, acceleration, force (including weight), momentum, impulse and electric field intensity. The resultant of two or more vectors is the single vector that has the same effect as all of them acting together. The equilibrant is the single vector that would cancel them out. It has the same magnitude as the resultant but acts in exactly the opposite direction.
Adding and resolving vectors
A vector is drawn as an arrow. Its length, to a chosen scale, shows the magnitude, and its arrowhead shows the direction. Triangle law: draw the first vector, then draw the second starting from the head of the first (head to tail). The resultant is the arrow drawn from the tail of the first vector to the head of the second. Parallelogram law: draw both vectors from the same point (tail to tail) and complete the parallelogram. The diagonal from that common point is the resultant. The two laws always agree; they are two ways of drawing the same idea. When the vectors act along the same line, add them if they point the same way and subtract if they point in opposite ways. A force of 6 N east and a force of 2 N west give a resultant of 4 N east. When the two vectors are at right angles, Pythagoras' theorem gives the size of the resultant and the tangent ratio gives its direction. For any other angle, use the cosine-rule form in the formula box, or resolve both vectors into components and add the components. Resolving is the reverse of adding. A single force F acting at an angle θ to the horizontal has exactly the same effect as two perpendicular forces: a horizontal component F cos θ and a vertical component F sin θ. This is why a box pulled with a rope at an angle is dragged forward by only part of the pull.
Formulae to know
Perpendicular vectors: R = √(F₁² + F₂²), with direction tan θ = F₂ ÷ F₁ (θ measured from F₁) Two vectors with an angle θ between them: R² = F₁² + F₂² + 2F₁F₂ cos θ Components of F at angle θ to the horizontal: Fx = F cos θ (horizontal) and Fy = F sin θ (vertical) Resultant from components: R = √((ΣFx)² + (ΣFy)²), with direction tan α = ΣFy ÷ ΣFx Range of the resultant of two vectors: (F₁ − F₂) ≤ R ≤ (F₁ + F₂)
Worked example 1: forces at right angles
Two forces of 3 N (due east) and 4 N (due north) act on a body. Find the magnitude and direction of their resultant.
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: resolving a pull
A boy drags a loaded box along level ground with a rope inclined at 30° to the horizontal. The tension in the rope is 50 N. Find the horizontal and vertical components of the pull.
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): forces at 60°
Two forces of 5 N and 8 N act at a point, with an angle of 60° between them. Find the magnitude of their resultant.
Active Worked Example · Predict Step 1
6 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Using cos for the vertical component out of habit. The component next to the angle uses cos; the one opposite it uses sin. If the angle is given from the vertical, the vertical component becomes F cos θ. • Adding magnitudes like ordinary numbers. 3 N + 4 N = 7 N only when the forces act in the same direction. • Using the wrong angle in the cosine formula. In R² = F₁² + F₂² + 2F₁F₂ cos θ, θ is the angle between the vectors drawn tail to tail. Some textbooks write R² = F₁² + F₂² − 2F₁F₂ cos α, where α is the angle inside the vector triangle (α = 180° − θ). Both give the same answer if you use the matching angle. • Treating weight as a scalar. Mass is a scalar, but weight is a force, so it is a vector that always acts vertically downwards.
Summary
• Scalars have magnitude only (mass, speed, energy); vectors have magnitude and direction (displacement, velocity, force, momentum). • Add vectors head to tail (triangle law) or tail to tail (parallelogram law); the resultant is the single equivalent vector. • Perpendicular vectors: R = √(F₁² + F₂²) and tan θ = F₂ ÷ F₁. Any angle: R² = F₁² + F₂² + 2F₁F₂ cos θ. • Resolve a force F at θ to the horizontal into F cos θ (horizontal) and F sin θ (vertical). • The resultant always lies between F₁ − F₂ and F₁ + F₂; the equilibrant is equal and opposite to the resultant.
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