Scalars and vectors: adding and resolving vectors: WASSCE Physics

Push a stalled car with a friend and the result depends not only on how hard you each push, but on which way. That "which way" is what separates vectors from scalars, and it runs through mechanics, electricity and fields.

Learning objectives

By the end of this lesson you should be able to:

distinguish scalar and vector quantities and give examples of each
represent a vector by a directed line drawn to scale
add vectors using the triangle and parallelogram laws, by drawing and by calculation
resolve a vector into two perpendicular components

Scalars and vectors

Definition

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

Scalars	Vectors
distance	displacement
speed	velocity
mass	weight (a force)
time	acceleration
energy, work, power	force
temperature	momentum, impulse
pressure	electric field strength
density, volume	magnetic flux density

Common mistake

Speed is a scalar; velocity is a vector. A keke going round a roundabout at a steady 20 km h⁻¹ has constant speed but changing velocity, because its direction keeps changing. Likewise distance is a scalar, displacement a vector.

Scalars add by ordinary arithmetic: 3 kg + 4 kg = 7 kg always. Vectors do not: a 3 N force and a 4 N force can give a resultant anywhere from 1 N to 7 N, depending on the angle between them.

Representing a vector

A vector is drawn as a straight line with an arrowhead:

the length of the line, drawn to a chosen scale, represents the magnitude
the direction of the arrow represents the direction of the vector

Example: with a scale of 1 cm to 10 N, a force of 45 N acting due east is a line 4.5 cm long pointing east.

The resultant of two vectors

Definition

The resultant of two or more vectors is the single vector that has the same effect as all of them acting together.

Vectors in the same or opposite directions

Same direction: add. 5 N east + 3 N east = 8 N east.
Opposite directions: subtract. 5 N east + 3 N west = 2 N east.

The parallelogram law

Two vectors P and Q drawn from a point O form two sides of a parallelogram; the diagonal from O is the resultant R

Figure 1. The parallelogram law. WASSCE Tutor diagram.

Definition

Parallelogram law of vectors: if two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from that point.

The triangle law is the same idea: draw the vectors nose to tail; the resultant joins the tail of the first to the nose of the last.

Calculating the resultant

Perpendicular vectors (90°): use Pythagoras and tangent.

R = \sqrt{P^2 + Q^2} \qquad \tan\theta = \dfrac{Q}{P}

\begin{gathered}R = \sqrt{P^2 + Q^2} \\[6pt] \tan\theta = \dfrac{Q}{P}\end{gathered}

Any angle θ between them: use the cosine rule form

R^2 = P^2 + Q^2 + 2PQ\cos\theta

\begin{aligned}R^2 &= P^2 + Q^2 + 2PQ\cos\theta\end{aligned}

Worked example

A boy walks 8 m east and then 6 m north. Find his displacement.

The two displacements are at 90°:

R = \sqrt{8^2 + 6^2} = \sqrt{100} = 10\ \text{m}

\begin{aligned}R &= \sqrt{8^2 + 6^2} \\ &= \sqrt{100} \\ &= 10\ \text{m}\end{aligned}

\tan\theta = \dfrac{6}{8} = 0.75 \Rightarrow \theta = 36.9°

\begin{gathered}\tan\theta = \dfrac{6}{8} = 0.75 \\[6pt] \Rightarrow \theta = 36.9°\end{gathered}

Displacement: 10 m in the direction N 53.1° E (that is, 36.9° north of east). The distance he walked is 14 m, but his displacement is only 10 m.

Worked example

Two forces of 5 N and 12 N act at a point with an angle of 60° between them. Find the resultant.

R^2 = 5^2 + 12^2 + 2(5)(12)\cos 60° = 25 + 144 + 60 = 229

\begin{aligned}R^2 &= 5^2 + 12^2 + 2(5)(12)\cos 60° \\ &= 25 + 144 + 60 \\ &= 229\end{aligned}

R = \sqrt{229} = 15.1\ \text{N}

Check it is sensible: it lies between $12 - 5 = 7$ N and $12 + 5 = 17$ N.

Resolving a vector into components

A single vector can be replaced by two perpendicular vectors, its components, that together have the same effect.

A force F at angle theta to the horizontal, with horizontal component F cos theta and vertical component F sin theta

Figure 2. Resolving a vector. WASSCE Tutor diagram.

For a vector $F$ at angle $\theta$ to the horizontal:

F_x = F\cos\theta \quad\text{(horizontal)} \qquad F_y = F\sin\theta \quad\text{(vertical)}

\begin{gathered}F_x = F\cos\theta \\[6pt] \text{(horizontal)} \\[6pt] F_y = F\sin\theta \\[6pt] \text{(vertical)}\end{gathered}

Worked example

A girl pulls a loaded basin across the floor with a rope at 30° to the horizontal. The tension in the rope is 40 N. Find the components.

Horizontal (pulls the basin along): $40\cos 30° = 40 \times 0.866 = 34.6$ N
Vertical (lifts it slightly): $40\sin 30° = 40 \times 0.5 = 20$ N

Only the horizontal 34.6 N moves the basin forward, which is why pulling at a steep angle wastes effort.

Common mistake

The cos goes with the component next to the angle (adjacent side) and sin with the one opposite it. If the angle is measured from the vertical, the vertical component is $F\cos\theta$ . Always check where the angle is measured from.

Key points

Scalars: magnitude only. Vectors: magnitude and direction.
Resultant by the parallelogram or triangle law.
Perpendicular vectors: $R = \sqrt{P^2 + Q^2}$ , $\tan\theta = Q/P$ .
General angle: $R^2 = P^2 + Q^2 + 2PQ\cos\theta$ .
Components: $F\cos\theta$ along the angle's arm, $F\sin\theta$ perpendicular to it.

WASSCE examination focus

WAEC tests this

Objective questions ask you to classify quantities as scalar or vector and to find the resultant of two perpendicular forces. Theory questions often ask you to state the parallelogram law, then find a resultant graphically (state your scale, draw accurately, measure length and angle) or by calculation. Always give both magnitude and direction.

Quick check

Which is a vector: speed, energy, momentum, mass?
Forces of 6 N and 8 N act at right angles. Find the resultant.
A force of 50 N acts at 60° to the horizontal. Find its horizontal component.

Answers: 1. Momentum. 2. $\sqrt{36 + 64} = 10$ N at $\tan^{-1}(8/6) = 53.1°$ to the 6 N force. 3. $50\cos 60° = 25$ N.

Scalars and vectors: adding and resolving vectors

Learning objectives

Scalars and vectors

Representing a vector

The resultant of two vectors

Vectors in the same or opposite directions

The parallelogram law

Calculating the resultant

Resolving a vector into components

Key points

WASSCE examination focus

Quick check

Keep going with scalars and vectors

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