Topic 1.1 Scalars and Vectors in One Dimension: distinguish scalar and vector quantities, and add and subtract vectors along a single dimension using a chosen sign convention.

What this topic is asking

The College Board (Topic 1.1) wants you to tell the difference between a scalar and a vector, and to add and subtract vectors along one line using a sign convention. This sounds simple, but it is the grammar of the whole course: every kinematics and dynamics calculation depends on choosing a positive direction and tracking signs correctly. AP Physics 1 keeps vector arithmetic to one dimension in this topic, so the skill is really about disciplined use of positive and negative signs.

Scalars versus vectors

The everyday word "speed" is a scalar: $30$ m/s tells you everything. "Velocity" is a vector: $30$ m/s east is a complete description, and $30$ m/s west is a different velocity even though the speeds are equal. Keeping these straight is the single most common source of sign errors in AP Physics 1.

Sign conventions in one dimension

Because the direction is carried by the sign, one-dimensional vector addition is just ordinary signed arithmetic. If you walk $+5$ m (right) and then $-2$ m (left), your displacement is $+5 + (-2) = +3$ m, that is, $3$ m to the right. The magnitude of a vector is its size without the sign, written with absolute-value bars, so $|-3\ \text{m}| = 3$ m.

Adding and subtracting one-dimensional vectors

Two vectors along the same line either reinforce or oppose each other:

• Same direction: the signs match, so the magnitudes add. A $4$ N force right plus a $3$ N force right gives $+7$ N (right).
• Opposite directions: the signs differ, so the magnitudes partly cancel. A $4$ N force right plus a $3$ N force left gives $+4 + (-3) = +1$ N (right).

Subtraction is addition of the reverse vector: to find a change such as $\Delta v = v_f - v_i$, flip the sign of $v_i$ and add. This matters most when a velocity reverses direction, because then $v_f$ and $v_i$ have opposite signs and the change is larger than either value alone.

Distance and speed behave differently from displacement and velocity precisely because they are scalars. When you reverse direction, distance keeps accumulating (it can only increase), but displacement can shrink back toward zero. A runner doing one lap of a track covers a large distance but returns to the start, so the displacement for the lap is zero. This is why the average speed over a journey is always at least as large as the magnitude of the average velocity, and the two are equal only when the motion never reverses direction.

Why the distinction drives the whole course

Every quantity you meet in AP Physics 1 is either a scalar or a vector, and the rules for combining them differ. Forces (vectors) are added with attention to direction to get a net force; kinetic energy and time (scalars) are added as plain numbers. When you build a free-body diagram, draw a motion graph, or apply Newton's second law, you are really committing to a sign convention and then doing signed vector arithmetic. Getting this habit automatic now, on one-dimensional problems, means the two-dimensional vectors in Topic 1.5 and the force vectors in Unit 2 will feel like extensions of the same idea rather than something new. The examiners reward students who state their positive direction explicitly and then never break it.

Try this

Q1. A bird flies $20$ m north, then $12$ m south. Calculate its displacement and its distance travelled. [2 points]

• Cue. Take north as positive: displacement $= +20 - 12 = +8$ m (north); distance $= 20 + 12 = 32$ m.

Q2. Identify which of these are vectors: speed, force, mass, acceleration. [1 point]

• Cue. Force and acceleration are vectors; speed and mass are scalars.