1.4 Vectors and Scalar Quantities

Vector and Scalar Quantities

Physics describes the world using two types of quantities: scalars and vectors. Scalars only have a size (magnitude), while vectors carry both a size and a direction. Understanding the difference, and knowing how to work with vectors mathematically, is essential for nearly every topic you'll encounter in this course.

Scalar vs. Vector Quantities

A scalar is any quantity fully described by its magnitude alone. There's no direction involved.

• Mass (5 kg)
• Temperature (98.6°F)
• Time (3.2 s)
• Speed (60 mph)
• Energy (200 J)

A vector is a quantity that requires both magnitude and direction to be fully described.

• Displacement (10 m east)
• Velocity (25 m/s at 30° north of east)
• Acceleration (9.8 m/s² downward)
• Force (50 N to the right)
• Momentum (12 kg·m/s at 45°)

A common point of confusion: speed is a scalar, but velocity is a vector. Speed tells you how fast something moves; velocity tells you how fast and in what direction. The same distinction applies to distance (scalar) vs. displacement (vector).

Vector Operations and Calculations

Vector Addition combines two or more vectors into a single resultant vector $\vec{R} = \vec{A} + \vec{B}$. You can do this two ways:

• Graphically (tip-to-tail method): Draw the first vector, then place the tail of the second vector at the tip of the first. The resultant points from the tail of the first to the tip of the last.
• Analytically (component method): Add the corresponding components: $R_x = A_x + B_x$, $R_y = A_y + B_y$. This is the method you'll use most in problem-solving.

Vector Subtraction works by adding the negative of a vector. Flipping a vector's direction gives you its negative:

$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Scalar Multiplication scales a vector's magnitude and can reverse its direction. Multiplying $\vec{A}$ by a scalar $c$:

$c\vec{A} = (cA_x,\; cA_y,\; cA_z)$

If $c > 0$, the direction stays the same. If $c < 0$, the vector flips direction.

Vector Magnitude and Direction

To find the magnitude (length) of a vector from its components:

• In two dimensions: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$
• In three dimensions: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$

To find the direction angle measured from the positive x-axis:

$\theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$

Be careful with this formula. Your calculator's inverse tangent only returns values between $-90°$ and $90°$, so you need to check which quadrant the vector actually points into and adjust accordingly. If $A_x < 0$, add $180°$ to the calculator's result.

A unit vector points in the same direction as the original vector but has a magnitude of exactly 1. You calculate it by dividing the vector by its magnitude:

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

The standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ point along the x, y, and z axes, respectively.

Vector Components and Analysis

Resolving a vector means breaking it into perpendicular components along the x and y axes. If a vector has magnitude $A$ and makes an angle $\theta$ with the positive x-axis:

• x-component: $A_x = A\cos\theta$
• y-component: $A_y = A\sin\theta$

This is one of the most-used skills in introductory physics. You'll apply it to:

• Projectile motion: Breaking initial velocity into horizontal and vertical components
• Forces on inclined planes: Splitting gravity into components parallel and perpendicular to the surface
• Relative motion: Adding velocity vectors of objects moving in different directions

Dot Product (Scalar Product) multiplies two vectors and returns a scalar:

$\vec{A} \cdot \vec{B} = AB\cos\theta$

where $\theta$ is the angle between the two vectors. This shows up in work calculations, where $W = \vec{F} \cdot \vec{d}$. When the vectors are perpendicular ($\theta = 90°$), the dot product is zero.

Cross Product (Vector Product) multiplies two vectors and returns a new vector:

$|\vec{A} \times \vec{B}| = AB\sin\theta$

The direction of the resulting vector is perpendicular to both $\vec{A}$ and $\vec{B}$, determined by the right-hand rule: point your fingers along $\vec{A}$, curl them toward $\vec{B}$, and your thumb points in the direction of $\vec{A} \times \vec{B}$. Cross products appear in torque ($\vec{\tau} = \vec{r} \times \vec{F}$) and magnetic force calculations. When the vectors are parallel ($\theta = 0°$), the cross product is zero.