Topic 1.1 Notes – Scalars and Vectors

1. What Scalars and Vectors Are

Scalars

A scalar is fully described by a magnitude only (number + units).

Examples you already use:

• Mass (2 kg)
• Time (4 s)
• Energy (10 J)
• Distance (total path length)
• Speed (how fast, no direction)

Scalars follow normal algebra. If you walk 3 m and then 2 m, your total distance is 5 m. No direction bookkeeping.

Vectors

A vector needs both magnitude and direction.

Examples:

• Position
• Displacement
• Velocity
• Acceleration
• Force, momentum

If direction affects the physics, it’s a vector.

Key scalar-vector pairs (know these cold)

• Distance (scalar) vs Displacement (vector)
  Distance = total path.
  Displacement = straight-line change in position + direction.
• Speed (scalar) vs Velocity (vector)
  Speed = how fast.
  Velocity = how fast and which way.

On quizzes, they love giving a motion description and asking which quantities could be zero. A runner finishing a lap has nonzero distance but zero displacement.

Direction in One Dimension

In 1D, we don’t draw arrows. We use sign.

1. Choose a positive direction (you must do this first).
2. Anything opposite that direction gets a negative sign.

If +x is right, then motion left has negative velocity. That sign replaces the arrow.

2. How Vectors Are Represented

Arrow Representation

Geometrically, a vector is an arrow.

Vector shown as an arrow in the x-y plane

• Length ∝ magnitude
• Arrowhead = direction
• Tail → starting point
• Tip → ending point

In the diagram, the red arrow starts at the origin and points into the first quadrant. Its length represents how large the vector is, and its orientation shows the direction.

You can slide a vector parallel to itself without changing it. Its location in space doesn’t matter, only its length and direction. That idea becomes important when adding forces.

Magnitude and Direction Form

Sometimes a vector is written like:

• 12 m/s at 40° above +x
• 25 N east

In 2D, convert to components using trig:

$$A_{x} = A\cos\theta, \quad A_{y} = A\sin\theta$$

Angle is usually measured from +x unless stated otherwise. If they say “30° north of west,” slow down and sketch it.

Unit Vector Notation

In AP Physics C, this is the default language.

Standard unit vectors:

• $\hat{i}$ → +x
• $\hat{j}$ → +y
• $\hat{k}$ → +z

Each has magnitude 1.

General form:

$$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$$

Magnitude:

$$|\vec{A}| = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$$

If $\vec{A} = 3\hat{i} - 4\hat{j}$, its magnitude is 5. The negative sign tells you it points in −y.

You’ll constantly move between component form and magnitude-direction form.

Position Vector and Radial Unit Vector

The position vector $\vec{r}$ points from the origin to a point.

If a particle is at (x, y, z):

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

The radial unit vector:

$$\hat{r} = \frac{\vec{r}}{|\vec{r}|}$$

It has magnitude 1 and points in the same direction as $\vec{r}$. You’ll use this later for gravitational and other central forces.

3. Vector Components and Resultant Vectors

Components

Any 2D vector can be written:

$$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$$

• $A_{x}$ = how much it points along x
• $A_{y}$ = how much along y
• Sign tells direction along that axis

Once broken into components, vectors behave like scalars in each direction separately.

Resultant Vectors

A resultant vector is just the vector sum.

Add component by component:

$$R_{x} = A_{x} + B_{x}, \quad R_{y} = A_{y} + B_{y}$$

Then:

$$\vec{R} = R_{x}\hat{i} + R_{y}\hat{j}$$

Magnitude and direction:

$$|\vec{R}| = \sqrt{R_{x}^{2} + R_{y}^{2}}, \quad \theta = \tan^{-1}\left(\frac{R_{y}}{R_{x}}\right)$$

On FRQs, this is how you combine forces before applying Newton’s Second Law.

4. Thinking About Scalars and Vectors in Mechanics

Direction check

Before writing an equation, ask yourself whether direction matters.

• If yes → use components or full vector notation.
• If no → scalar treatment is fine.

1D Motion

In one dimension:

$$v_{x} = v_{x0} + a_{x} t$$

Everything has a subscript x. The sign handles direction. A negative acceleration just means opposite your chosen positive axis.

Students lose points by mixing sign conventions halfway through a problem. Choose once. Stick with it.

Why This Matters

Newton’s Second Law is a vector equation:

$$\sum \vec{F} = m\vec{a}$$

In practice, you split it into x, y (and maybe z) equations. Every motion problem you solve this year depends on clean vector thinking.

If you’re comfortable moving between:

• Arrow diagrams
• Components
• Magnitude + direction

you’re set up for the rest of Mechanics.