Introduction to Vector Analysis in Physics and Engineering

Resolving a vector means finding its projection along the $x$ and $y$ axes. If we have a vector $\vec{A}$ with a magnitude $A$ and an angle $\theta$ measured from the positive $x$-axis, we can use trigonometry to find its components:

The vector can then be written in unit vector notation, which uses $\hat{i}$ to represent the $x$-direction and $\hat{j}$ to represent the $y$-direction:

Worked Example: Resolving a Force

Suppose a person pulls a sled with a force of $50 \text{ N}$ at an angle of $30^\circ$ above the horizontal. To find the horizontal and vertical components of this force:

1. Identify the magnitude and angle: $F = 50 \text{ N}$, $\theta = 30^\circ$.
2. Calculate $F_x$:

3. Calculate $F_y$:

The force vector is $\vec{F} = 43.3\hat{i} + 25\hat{j} \text{ N}$.

Vector Multiplication: Dot and Cross Products

Vectors cannot be multiplied like ordinary numbers. Depending on the physical context, we use two distinct types of multiplication: the dot product and the cross product.

The Dot Product (Scalar Product)

The dot product takes two vectors and returns a scalar. It measures how much one vector "lines up" with another. For two vectors $\vec{A}$ and $\vec{B}$ with an angle $\theta$ between them, the dot product is defined as:

In terms of components, it is calculated as:

Physical Application: The dot product is used to calculate Work. Work is the dot product of the Force vector and the Displacement vector ($W = \vec{F} \cdot \vec{d}$). If the force is perpendicular to the displacement, the dot product is zero, meaning no work is done.

The Cross Product (Vector Product)

Unlike the dot product, the cross product takes two vectors and returns a new vector. This new vector is unique because it is perpendicular to the plane formed by the original two vectors. The magnitude of the cross product is:

The direction of the resulting vector is determined by the Right-Hand Rule: if you curl the fingers of your right hand from $\vec{A}$ toward $\vec{B}$, your thumb points in the direction of $\vec{A} \times \vec{B}$.

Physical Application: The cross product is used to calculate Torque ($\vec{\tau} = \vec{r} \times \vec{F}$), which describes the rotational tendency of a force applied at a distance from a pivot point.

Summary Table: Dot vs. Cross Product

By mastering these fundamentals—distinguishing types, resolving components, and applying the correct product—you build the mathematical toolkit necessary to navigate the complexities of the physical world.