Scalars and vectors in one dimension

Two moves underlie this topic: recognize whether a quantity is a scalar (magnitude only) or a vector (magnitude plus direction), and in 1D read direction from the sign. Every equation from Topic 1.2 on depends on getting these right.

A scalar is a quantity described by magnitude only: a number and a unit, with no direction attached. A vector is described by both a magnitude AND a direction.

The canonical examples for Unit 1:

• Scalarsmass, time, distance, speed, temperature.
• Vectorsposition, displacement, velocity, acceleration, force.

A vector is drawn as an arrow whose length is proportional to its magnitude and whose direction shows its direction. Twice the magnitude, twice the length.

Three pieces of notation worth fixing right now:

• $\vec v$ denotes the full velocity vector (magnitude AND direction together).
• $|\vec v|$ denotes the magnitude of $\vec v$, also called the speed. This is a scalar.
• $v$ or $v_x$ (no arrow) denotes the component along a chosen axis. In 1D, the sign of $v_x$ encodes the direction, so the arrow notation is dropped.

The magnitude of a velocity is a speed. Same pattern for distance/displacement and force-magnitude/force-vector: the scalar is the vector with direction stripped off.

The pivotal idea. In 1D you don't draw arrows; you pick a positive direction once and write every vector as a signed number along that axis. With "east is positive" (the convention used throughout this lesson):

• A vector pointing east gets a positive sign.
• A vector pointing west gets a negative sign.
• The magnitude is what you get when you strip the sign: $|v| = $ "how big the vector is, regardless of which way it points."

Worked example. A bird flies due west at $6$ m/s. East is positive. What is the bird's velocity?

Two vectors along the same 1D axis add component-wise:

Sum the signed components. The result is a signed component along the same axis: its size is the magnitude, its sign is the direction.

Worked example. Two pulls act on the same point of a cart, both along the east-west axis. Pull A is $+12$ N (east); pull B is $-7$ N (west). What is the net force?

Three habits that prevent silly errors:

1. Don't add magnitudes when the signs differ.
   $12 + 7 = 19$ N would be the answer if both pulls pointed east. Two pulls in opposite directions don't add their magnitudes; they add their signed components.
2. Don't collapse the sum into a magnitude subtraction.
   Writing $|12 - 7| = 5$ stripped of sign loses the direction. The signed sum keeps both magnitude AND direction; treating it as scalar subtraction throws the sign away.
3. Use parentheses around negative numbers.
   $12 + -7$ written without grouping invites mid-step confusion. Parentheses keep the structure visible: $(+12) + (-7) = +5$.

Three flavors of sign confusion that recur on Unit 1 questions.

Sign as magnitude. A particle at $v = -15$ m/s is moving FASTER than one at $v = +8$ m/s. The sign is direction; the size is $15 > 8$. Reading "more negative" as "smaller" causes about half the mistakes on this topic.

Sign as state. A particle at $v = -8$ m/s is not at rest, not slowing, not impossible — it's moving at $8$ m/s in the negative direction. The sign is geometric (which way), not metaphysical.

Coordinate / convention error. Positive direction is your choice. "Positive equals forward" only when you've chosen forward as positive. If the problem sets a convention, use it; if not, set one and apply it to every vector in the problem.

The sign is information about direction along the chosen axis. It is not size, it is not state, and it is not absolute.

Three pitfalls that map to the diagnostic's misconception codes.