Distinguish between scalar and vector quantities, including identifying examples and applying operations of addition and subtraction in one and two dimensions

What this dot point is asking

QCAA wants you to separate the two families of physical quantity. Scalars are fully described by a magnitude and a unit. Vectors need a magnitude, a unit, and a direction. The same arithmetic rules do not apply to both. This distinction underpins every motion, force and field problem in Units 2, 3 and 4.

Scalars and vectors

Two scalars combine by ordinary arithmetic. Two vectors combine by vector addition, which depends on their directions.

Adding vectors graphically

Place the tail of the second vector at the head of the first. The resultant runs from the tail of the first to the head of the last. The order does not matter (vector addition is commutative).

For perpendicular vectors, magnitude follows Pythagoras and direction follows the inverse tangent. For non-perpendicular vectors, resolve into components or use the cosine rule.

Resolving into components

A vector $\vec{v}$ of magnitude $v$ at angle $\theta$ above the horizontal has components:

Components allow you to add and subtract vectors numerically. Sum the $x$-components, sum the $y$-components, then recombine with Pythagoras and arctan.

Subtracting vectors

$\vec{a} - \vec{b}$ means $\vec{a} + (-\vec{b})$. To subtract $\vec{b}$, reverse its direction and add. This step is critical for change-in-velocity problems (for example $\Delta \vec{v} = \vec{v}_f - \vec{v}_i$ in uniform circular motion or in a collision).

Worked example

A car is travelling east at $20$ m s$^{-1}$ then turns and travels north at $20$ m s$^{-1}$. Find the change in velocity.

$\vec{v}_i = 20\hat{\imath}$ (east), $\vec{v}_f = 20\hat{\jmath}$ (north).

$\Delta \vec{v} = \vec{v}_f - \vec{v}_i = -20\hat{\imath} + 20\hat{\jmath}$.

Magnitude: $|\Delta \vec{v}| = \sqrt{20^2 + 20^2} = 28.3$ m s$^{-1}$.

Direction: $45°$ north of west.

The speed did not change but the velocity did. This is exactly the source of centripetal acceleration in circular motion, which you will meet in Unit 3.

Common traps

Calling speed and velocity the same thing. Speed is the magnitude of velocity. They have the same SI unit but velocity is a vector. A constant-speed object turning a corner has changing velocity.

Adding magnitudes when directions differ. A $3$ m walk east followed by $4$ m north is a $7$ m total distance but a $5$ m displacement. Magnitudes do not add unless the vectors are collinear.

Forgetting to specify a reference for direction. "Magnitude $5$ N at $30°$" is ambiguous. State the reference axis ($30°$ above the horizontal, $30°$ east of north).

Mixing degrees and radians on a calculator. Trig functions return different values depending on calculator mode. Set degrees for QCAA Physics unless a problem explicitly uses radians.

In one sentence

A scalar has only magnitude (mass, time, distance, speed, energy); a vector has magnitude and direction (displacement, velocity, acceleration, force, momentum) and is added or subtracted graphically (head to tail) or by resolving into perpendicular components.