Essential Vector Operations

Why This Matters

Vectors are the language of motion, force, and space. You'll encounter them constantly throughout calculus and beyond. When you work with position, velocity, and acceleration in multivariable calculus, or analyze work, torque, and flux in physics, you're using vector operations. In this course, the goal is understanding how vectors combine, how they measure relationships between directions, and how they describe geometric objects in space.

Don't just memorize formulas. Know what each operation does conceptually. Can you explain why the dot product tells you about angles? Why the cross product gives you area? Master the underlying geometry, and the calculations become intuitive.

---

Combining and Scaling Vectors

These foundational operations let you build new vectors from existing ones. Addition combines effects; scalar multiplication stretches or shrinks without changing direction (unless you flip it).

Vector Addition and Subtraction

• Component-wise combination: add corresponding components: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$
• Geometric interpretation uses the parallelogram law or tip-to-tail method to visualize the resultant
• Subtraction means adding the negative: $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$, which geometrically points from $\vec{v}$'s tip to $\vec{u}$'s tip when both vectors start at the origin

Scalar Multiplication

• Scaling magnitude: multiplying by scalar $c$ gives $c\vec{v}$ with magnitude $|c||\vec{v}|$
• Direction preserved when $c > 0$; direction reversed when $c < 0$
• Zero scalar produces the zero vector regardless of the original vector's magnitude

---

Measuring Vectors

Before you can work with vectors effectively, you need to quantify their size and standardize their direction. Magnitude gives length; unit vectors give pure direction.

Magnitude (Length) of a Vector

• Pythagorean extension: for $\vec{v} = \langle x, y, z \rangle$, magnitude is $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$
• Always non-negative: magnitude equals zero only for the zero vector
• Distance formula connection: the magnitude of $\vec{PQ}$ gives the distance between points $P$ and $Q$

For a 2D vector, just drop the $z$-component: $|\vec{v}| = \sqrt{x^2 + y^2}$. The idea is the same either way.

Unit Vectors

• Magnitude of exactly 1: obtained by dividing any nonzero vector by its magnitude: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$
• Standard basis vectors $\hat{i}, \hat{j}, \hat{k}$ point along the positive $x$-, $y$-, and $z$-axes respectively
• Direction isolation: unit vectors let you separate how far from which way

For example, if $\vec{v} = \langle 3, 4 \rangle$, then $|\vec{v}| = 5$ and $\hat{v} = \langle \frac{3}{5}, \frac{4}{5} \rangle$. You've kept the direction but scaled the length to 1.

---

Products That Reveal Geometric Relationships

The dot and cross products are your primary tools for measuring angles, checking perpendicularity, and finding areas. The dot product outputs a scalar; the cross product outputs a vector. Keeping that distinction straight is half the battle.

Dot Product (Scalar Product)

• Two equivalent formulas: $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3$ (component form) or $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos\theta$ (geometric form)
• Angle finder: rearrange to get $\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}$
• Orthogonality test: vectors are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$

The sign of the dot product tells you something useful: positive means the angle between the vectors is acute (less than 90°), zero means exactly 90°, and negative means obtuse (greater than 90°).

Cross Product (Vector Product)

• Produces a perpendicular vector: $\vec{u} \times \vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$
• Determinant calculation: $\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$
• Magnitude equals parallelogram area: $|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin\theta$
• Order matters: $\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})$. The cross product is not commutative. Use the right-hand rule to determine the direction of the resulting vector.

Note that the cross product is only defined for 3D vectors. The dot product works in any dimension.

---

Breaking Vectors Apart

Sometimes you need to analyze how much of a vector points in a particular direction. Projection extracts the component along another vector; decomposition separates into coordinate directions.

Vector Projection

• Formula: $\text{proj}_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\vec{v}$ gives the component of $\vec{u}$ in the direction of $\vec{v}$
• Scalar projection (signed length along $\vec{v}$) is $\text{comp}_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|}$
• Physics application: resolving force into components parallel and perpendicular to a surface

Watch the difference: the scalar projection gives you a number (positive or negative), while the vector projection gives you an actual vector pointing along $\vec{v}$.

Vector Decomposition

• Component form: any vector $\vec{v} = \langle a, b, c \rangle$ decomposes as $a\hat{i} + b\hat{j} + c\hat{k}$
• Arbitrary directions: you can decompose along any set of basis vectors, not just coordinate axes
• Problem-solving strategy: choose axes aligned with the geometry of your problem to simplify calculations

---

Describing Geometric Objects

Vectors provide elegant equations for lines and planes in 3D space. A line needs a point and a direction; a plane needs a point and a normal.

Vector Equations of Lines and Planes

• Line equation: $\vec{r}(t) = \vec{r}_0 + t\vec{v}$ where $\vec{r}_0$ is a position vector to a known point and $\vec{v}$ is the direction vector
• Plane equation: $\vec{n} \cdot (\vec{r} - \vec{r}_0) = 0$ where $\vec{n}$ is the normal vector (perpendicular to the plane)
• Parametric conversion: vector equations convert directly to parametric form by writing out each component separately

For the line, as $t$ varies over all real numbers, the point $\vec{r}(t)$ traces out every point on the line. Each value of $t$ gives you a different point.

Vector-Valued Functions

• Maps scalars to vectors: $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ traces a curve as $t$ varies
• Derivatives give velocity: $\vec{r}'(t)$ is tangent to the curve and represents instantaneous velocity
• Integrals recover position: integrating acceleration gives velocity; integrating velocity gives position

---

Quick Reference Table

---

Self-Check Questions

1. Which two operations both produce scalar outputs, and how do their geometric interpretations differ?

2. If $\vec{u} \cdot \vec{v} = 0$ and $\vec{u} \times \vec{v} = \vec{0}$, what can you conclude about vectors $\vec{u}$ and $\vec{v}$?

3. How would you use the dot product versus the cross product to find the angle between two vectors? What's the practical difference?

4. A vector-valued function $\vec{r}(t)$ describes a particle's position. What operation gives the particle's speed at time $t$, and what combination of concepts does this require?

5. You're given vectors $\vec{u}$ and $\vec{v}$ and asked for the area of the triangle they form. Which operation do you use, and what adjustment must you make to get triangle area instead of parallelogram area?