Scalars and Vectors: Adding Vectors and Finding Resultants

By Engr. Henry Akpan

Size, or Size and Direction?

If you walk 3 km and then 4 km, how far are you from where you started? It could be anywhere from 1 km to 7 km, depending on the directions you walked. Quantities like this, where direction matters, are vectors. Quantities that only have a size, such as mass or temperature, are scalars. Understanding scalars and vectors is the first step in mechanics, and it underlies forces, velocities, fields and almost every calculation in physics. This tutorial explains the difference, how to add and subtract vectors using the triangle and parallelogram rules, how to find resultants by calculation and scale drawing, and how vectors are written in component form, with WAEC and JAMB-style examples.

Scalars and Vectors Defined

A scalar quantity has magnitude (size) only. A vector quantity has both magnitude and direction.

• Scalars: distance, speed, mass, time, energy, work, power, temperature, density, volume, electric charge, potential.
• Vectors: displacement, velocity, acceleration, force, weight, momentum, impulse, electric field strength, magnetic flux density.

Pairs of scalars and vectors to watch: distance is a scalar, but displacement (distance in a stated direction) is a vector; speed is a scalar, but velocity is a vector. A car going round a roundabout at a steady 30 km/h has constant speed but changing velocity, because its direction keeps changing.

Some quantities look like vectors but are not. Work is force times displacement, both vectors, but work itself is a scalar (it is a dot product). Electric current has a direction along a wire, but it does not add like a vector, so it is treated as a scalar.

Representing a Vector

A vector is drawn as an arrow. The length of the arrow (to a chosen scale) shows the magnitude, and the arrow points in the direction. In print, vectors are written in bold (F) or with an arrow over the letter, and the magnitude is written F or |F|.

Directions are given as angles from a reference line, or as compass bearings, measured clockwise from north (for example, a bearing of 060° is 60° east of north).

Adding Vectors: The Triangle Rule

To add two vectors A and B, draw A, then draw B starting from the head (tip) of A. The resultant A + B is the arrow from the tail of A to the head of B. This is the "head to tail" or triangle rule.

For more than two vectors, keep joining head to tail; the resultant closes the polygon from the first tail to the last head. If the vectors form a closed shape, the resultant is zero, which is the condition for equilibrium of forces.

Adding Vectors: The Parallelogram Rule

Alternatively, draw both vectors from the same point and complete the parallelogram. The resultant is the diagonal from the common point. The parallelogram rule and the triangle rule always give the same answer.

Calculating the Resultant

Same direction: add the magnitudes. Forces of 5 N and 3 N pulling east give 8 N east.

Opposite directions: subtract, and the resultant is in the direction of the larger. 5 N east and 3 N west give 2 N east.

At right angles: use Pythagoras for the size and trigonometry for the direction.

Example 1. A boat is driven north at 4 m/s through water flowing east at 3 m/s. Find its resultant velocity.

Magnitude = √(4² + 3²) = √25 = 5 m/s. Direction: tan θ = 3/4, so θ = 36.9° east of north (a bearing of about 037°).

At any angle: for two vectors of sizes P and Q with an angle θ between them (drawn tail to tail), the cosine rule gives the resultant:

R² = P² + Q² + 2PQ cos θ.

(The plus sign appears because the angle inside the triangle is 180° − θ, and cos(180° − θ) = −cos θ.)

Example 2. Two forces of 6 N and 8 N act at a point with an angle of 60° between them. Find the resultant.

R² = 36 + 64 + 2 × 6 × 8 × cos 60° = 100 + 48 = 148, so R = √148 ≈ 12.2 N.

Its direction, using the sine rule in the triangle of vectors: the angle α between the resultant and the 6 N force satisfies sin α / 8 = sin 120° / 12.17, so sin α = 8 × 0.866/12.17 = 0.569 and α ≈ 34.7°.

Check the extremes: with θ = 0°, R = 6 + 8 = 14 N; with θ = 180°, R = 8 − 6 = 2 N. The resultant of two forces always lies between their difference and their sum.

Scale Drawing

In practical examinations you may be asked to find a resultant by scale drawing. Choose a scale (for example 1 cm to 1 N), draw the first vector accurately with a ruler and protractor, draw the second from its head, and measure the resultant's length and angle. Example 2 drawn at 1 cm to 1 N gives a resultant about 12.2 cm long. A larger scale gives a more accurate answer.

Subtracting Vectors

A − B means A + (−B), where −B has the same size as B but the opposite direction. Vector subtraction gives a change in a vector quantity, such as change in velocity.

Example 3. A ball moving east at 10 m/s bounces off a wall and moves west at 8 m/s. What is its change in velocity?

Taking east as positive: Δv = final − initial = (−8) − (+10) = −18 m/s, that is 18 m/s towards the west. Its change in speed is only 2 m/s, but its change in velocity is 18 m/s, which is why bouncing involves a large force.

Example 4. A car's velocity changes from 20 m/s north to 20 m/s east. Find the change in velocity.

Δv = v(east) − v(north). The two vectors are at right angles, so |Δv| = √(20² + 20²) = 28.3 m/s, directed south-east (a bearing of 135°). The speed did not change, but the velocity did.

Vectors in Component Form

Any vector can be split into two perpendicular components, usually along the x and y axes. A vector of magnitude F at angle θ to the x-axis has components

Fx = F cos θ and Fy = F sin θ.

Using unit vectors i (along x) and j (along y), the vector is written F = Fx i + Fy j. Adding vectors is then just adding components.

Example 5. Add A = 3i + 4j and B = 5i − 2j.

A + B = 8i + 2j. Magnitude = √(64 + 4) = √68 ≈ 8.25; direction tan⁻¹(2/8) ≈ 14.0° above the x-axis.

The detailed use of components to solve force problems on slopes and in equilibrium is covered in the tutorial on resolution of forces.

Multiplying Vectors by Scalars

Multiplying a vector by a positive scalar changes its size but not its direction: 3v is three times as long as v. Multiplying by a negative scalar also reverses the direction. Many physics laws are of this form: F = ma (a scalar mass times a vector acceleration) and p = mv.

Practice Questions

1. Which of these are vectors: speed, momentum, energy, weight, pressure? Answer: momentum and weight.
2. A man walks 5 km east and then 12 km north. Find his displacement. Answer: 13 km on a bearing of 022.6° (tan⁻¹(5/12) east of north).
3. Find the resultant of forces of 10 N and 10 N at 120° to each other. Answer: R² = 100 + 100 + 200 cos 120° = 100, so R = 10 N.
4. What is the greatest and least possible resultant of forces 7 N and 4 N? Answer: 11 N and 3 N.

Summary

• Scalars and vectors differ in direction: scalars have magnitude only; vectors have magnitude and direction.
• Distance and speed are scalars; displacement and velocity are vectors.
• Add vectors head to tail (triangle rule) or with the parallelogram rule.
• For an angle θ between two vectors, R² = P² + Q² + 2PQ cos θ; at right angles this is Pythagoras.
• Subtracting vectors gives the change in a vector quantity; components make adding vectors simple.