Scalars and Vectors — Quantities with Direction

Some measurements are complete with a single number and a unit. "The water is 21 °C." "The bag weighs 1.2 kg." Nobody asks "21 °C in which direction?" — the question makes no sense. But other measurements are only half-finished without a direction. "The wind is 30 km/h" leaves out the most important thing for a sailor: which way? Physics sorts every quantity into these two boxes.

A scalar is a quantity that has magnitude only — a size with a unit. A vector is a quantity that has both magnitude and direction. The magnitude of a vector is always a positive number; the direction is extra, essential information.

Sorting the motion quantities

A scalar has magnitude only (distance, speed, time, mass). A vector has magnitude AND direction (displacement, velocity, acceleration, force). A vector's magnitude is always positive; the direction is separate, essential information.

Pause — copy the two definitions and the two example lists (scalars vs vectors) into your book before moving on.

We just saw that vectors carry a direction while scalars do not. That raises a question: does that direction actually change any answer, or is it just extra labelling? This card answers it → with the drone trip, where the scalar and the vector give genuinely different results.

Run the drone trip again. It flies 6 km out to a remote clinic, drops the samples, and flies 6 km back. Ask "how far did it travel?" and the answer is a scalar: 12 km of path. Ask "how far is it from base now?" and the answer is a vector: it is back where it started, so 0 km.

The path length (a distance, scalar) added up to 12 km because distance never decreases — every metre of travel counts. But the change in position (a displacement, vector) is zero, because the outbound trip east and the return trip west cancel. Cancellation is only possible because the two legs point in opposite directions — and only vectors carry direction.

Distance (scalar) is the total path length and only ever grows. Displacement (vector) is the change in position from start to finish and can be zero or negative because opposite directions cancel. Same trip, two different numbers.

Pause — copy the drone example and the rule "scalars accumulate, vectors can cancel" into your book before moving on.

We just saw that direction lets vectors cancel where scalars cannot. That raises a question: how do we record a vector on paper so its direction is unmistakable? This card answers it → we draw it as an arrow, where the length is the magnitude and the arrowhead is the direction.

A number alone cannot show "east" or "30° above the road". So we draw vectors as arrows. Everything about the arrow means something: how long you draw it, and which way it points.

An arrow representing a vector has two carefully controlled features:

• Length = magnitude. You pick a scale (e.g. 1 cm represents 10 m/s) and draw the arrow that many times long. A 30 m/s velocity is drawn three times longer than a 10 m/s velocity.
• Arrowhead = direction. The way the head points is the direction of the vector. The tail is the starting point; the head is where it points.

Notation: writing a vector down

In handwriting a vector symbol is written with an arrow above it, $\vec{v}$, or in bold in print, $\mathbf{v}$. Its magnitude (just the size) is written without the arrow, $v$ or $|\vec{v}|$. So $\vec{v}$ might be "20 m/s north" while $v = 20$ m/s is only the speed.

You always state a vector as magnitude + direction: "12 m east", "9.8 m/s² downwards", "40 N at 30° above horizontal".

A vector is drawn as an arrow: length = magnitude (to a chosen scale), arrowhead = direction. Symbol $\vec{v}$ (or bold $\mathbf{v}$); magnitude is $v$ or $|\vec{v}|$. Equal vectors share magnitude AND direction; the negative vector $-\vec{a}$ has equal length but opposite direction.

Pause — copy the arrow rule (length = magnitude, head = direction), the notation, and the equal/negative vector definitions into your book before moving on.

We just saw that direction is drawn with an arrowhead. That raises a question: when all the motion is along a single straight line, do we really need to draw arrows every time? This card answers it → no — in one dimension we replace "east/west" with a simple + and − sign convention.

When everything happens along one line — a car on a straight road, a lift in a shaft — there are only two possible directions. Drawing arrows is overkill. Instead we agree on a sign convention: pick one direction to be positive, and the opposite direction is automatically negative.

For example, choose east = positive (+). Then west is negative (−). A displacement of "5 m east" becomes $+5$ m, and "5 m west" becomes $-5$ m. The sign now is the direction — no arrow needed. (You may pick either direction as positive; you just have to state your choice and stick to it.)

In one dimension, choose one direction as positive (+); the opposite is negative (−). The sign then carries the direction: +5 m and −5 m are equal magnitudes pointing opposite ways. State your sign convention before you calculate.

Pause — copy the sign-convention rule and one worked example (e.g. east = +, so 5 m west = −5 m) into your book before moving on.

We just saw that signs replace arrows in one dimension. That raises a question: how do we combine two vectors — like two legs of a journey — once they each carry a sign? This card answers it → we simply add the signed numbers; the sign of the result gives the resultant's direction.

Two vectors on the same line are combined by adding their signed values. The single vector that results is called the resultant. Its sign tells you the direction; its size tells you the magnitude.

Take the convention east = +. A person walks 8 m east ($+8$ m) then 3 m west ($-3$ m). The resultant displacement is:

Subtracting a vector is the same as adding its negative: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. This is exactly how we will later find a change in velocity (for acceleration): subtract the start vector from the end vector.

To add vectors on a line, give each a sign from your convention, then add the signed numbers. The resultant's sign is its direction; its size is its magnitude. Subtracting a vector = adding its negative: $\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$. Always keep the sign on the final answer.

Pause — copy the four-step method and the worked example ($+12 + (-20) = -8$ m = 8 m west) into your book before moving on.