Scalars and Vectors: The Building Blocks of Physics

• Displacement
• Velocity
• Acceleration
• Force
• Momentum

Exam Tip: Always check if a question asks for "distance" (scalar) or "displacement" (vector). If a person walks in a complete circle and ends up where they started, their distance is the circumference of the circle, but their displacement is zero!

Vector Operations

Because vectors have direction, you cannot always add them like simple numbers. You must account for the way they are oriented in space.

Addition and Subtraction

The most common way to add vectors is the "head-to-tail" method. To add multiple vectors, you place the tail of the second vector at the head of the first. The resultant vector is the single arrow drawn from the very first tail to the very last head.

If several vectors point in the same direction, you can simply sum their magnitudes. However, if a vector points in the opposite direction, it is treated as a negative value during addition.

The Pythagorean Theorem

When two vectors are perpendicular (acting at a 90-degree angle to each other), we can find the magnitude of the resultant vector using the Pythagorean Theorem.

If vector $a$ and vector $b$ are the two component vectors, the magnitude of the resultant $c$ is:

$$\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$$ $$c = \sqrt{a^{2} + b^{2}}$$

To find the specific direction (the angle $\varphi$), you can use trigonometry, such as:

$$\tan \varphi = \frac{\text{opposite}}{\text{adjacent}}$$

Worked Example: Peter walks 11 km North and then 11 km East. What is his resultant displacement? Since North and East are at right angles, we use the Pythagorean Theorem:

$$\begin{aligned} c^{2} &= 11^{2} + 11^{2} \\ c^{2} &= 121 + 121 \\ c^{2} &= 242 \\ c &= \sqrt{242} \approx 15.6 \text{ km} \end{aligned}$$

The direction would be Northeast.

Scalar Multiplication

You can multiply a vector by a scalar, which changes the vector's magnitude.

• If the scalar is positive, the direction remains the same.
• If the scalar is negative, the vector's direction is reversed.

Graphical Relationships in Physics

In physics experiments, we often use graphs to visualize how different quantities interact.

Linear and Direct Relationships

A direct relationship occurs when two variables change in the same manner: as one increases, the other also increases. This is expressed as $y = kx$, where $k$ is the constant of proportionality. On a graph, this appears as a straight line passing through the origin $(0,0)$.

Inverse Relationships

In an inverse relationship, the variables change in opposite manners: as one increases, the other decreases. This is expressed as $y = \frac{k}{x}$. The graph of an inverse relationship is not a straight line, but a curve.

Pro-Tip: If you have an inverse relationship (like $a$ and $M$), plotting $a$ against the reciprocal of the variable ($1/M$) will often result in a straight line. This makes it much easier to calculate the constant of proportionality using the slope of that line.

Analyzing Slopes

The slope of a graph can represent a specific physical value. For example, in a graph of speed ($v$) versus time ($t$), the slope represents the rate of change of speed, which is acceleration.

By mastering the distinction between scalars and vectors and learning how to manipulate them mathematically, you gain the tools necessary to solve the most fundamental problems in physics.