Key Concepts of Vectors in 3D Space

Why This Matters

Vectors in 3D space form the foundation for everything you'll encounter in multivariable calculus—from describing curves and surfaces to calculating work, flux, and circulation in later chapters. When you're working with parametric curves, line integrals, or surface integrals, you're building directly on the vector operations and geometric representations covered here. The concepts of dot products, cross products, projections, and vector equations show up repeatedly in both computational problems and theoretical questions.

You're being tested on more than just formulas. Exam questions will ask you to interpret what a dot product of zero means geometrically, or to set up a vector equation for a line through two points. Don't just memorize the mechanics—know what concept each operation illustrates and when to apply it. If you understand why the cross product produces a perpendicular vector or how projections decompose forces, you'll handle FRQs with confidence.

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Foundational Vector Operations

These operations are the building blocks for everything else. Master them first, and the rest of the topic becomes straightforward. Component-wise operations preserve the algebraic structure that makes vectors so powerful.

Definition of Vectors in 3D Space

• Ordered triplet representation—a vector is written as $(x, y, z)$ or $\langle x, y, z \rangle$, indicating displacement from the origin
• Magnitude and direction distinguish vectors from scalars; magnitude is calculated as $|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}$
• Geometric visualization as arrows in 3D space helps you interpret operations like addition (tip-to-tail) and scalar multiplication (stretching or shrinking)

Vector Operations (Addition, Subtraction, Scalar Multiplication)

• Component-wise addition—for $\mathbf{A} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{B} = \langle b_1, b_2, b_3 \rangle$, we get $\mathbf{A} + \mathbf{B} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle$
• Scalar multiplication scales magnitude without changing direction: $k\mathbf{A} = \langle ka_1, ka_2, ka_3 \rangle$, where negative $k$ reverses direction
• Linear combinations of vectors (like $c_1\mathbf{A} + c_2\mathbf{B}$) form the basis for parametric representations of lines and planes

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Products That Reveal Geometry

The dot and cross products aren't just computational tools—they encode geometric information about angles, alignment, and perpendicularity. Understanding their geometric meaning is more important than memorizing formulas.

Dot Product and Its Properties

• Algebraic formula: $\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3$, yielding a scalar (not a vector)
• Geometric interpretation: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta$, so a dot product of zero means the vectors are perpendicular
• Key properties include commutativity ($\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$) and distributivity over addition—essential for simplifying expressions

Cross Product and Its Properties

• Result is a vector perpendicular to both inputs: $\mathbf{A} \times \mathbf{B}$ is orthogonal to the plane containing $\mathbf{A}$ and $\mathbf{B}$
• Magnitude formula: $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin\theta$, which equals the area of the parallelogram formed by the two vectors
• Anti-commutativity ($\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}$) and the right-hand rule determine direction—order matters!

Projections of Vectors

• Projection formula: $\text{proj}_{\mathbf{B}}\mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2}\mathbf{B}$, giving the component of $\mathbf{A}$ in the direction of $\mathbf{B}$
• Decomposition splits any vector into parallel and perpendicular components relative to another vector
• Physical applications include resolving forces, calculating work ($W = \mathbf{F} \cdot \mathbf{d}$), and finding distances to lines

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Equations of Lines and Planes

These representations let you describe geometric objects algebraically. The key insight: lines need one parameter, planes need two (or a normal vector constraint).

Vector Equations of Lines

• Standard form: $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$, where $\mathbf{r}_0$ is a point on the line and $\mathbf{v}$ is the direction vector
• Parameter $t$ ranges over all real numbers, tracing every point on the line as it varies
• Finding direction vectors between two points: if the line passes through $P$ and $Q$, use $\mathbf{v} = \overrightarrow{PQ}$

Parametric Equations of Lines

• Component form: $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$, where $(a, b, c)$ is the direction vector
• Symmetric equations eliminate $t$: $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$ (watch for zero denominators)
• Converting between forms is a common exam task—practice moving from vector to parametric to symmetric

Vector Equations of Planes

• Point-normal form: $\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$, where $\mathbf{n}$ is the normal vector perpendicular to the plane
• Scalar equation: $Ax + By + Cz = D$, where $\langle A, B, C \rangle$ is the normal vector
• Two-parameter form: $\mathbf{r}(s, t) = \mathbf{r}_0 + s\mathbf{A} + t\mathbf{B}$, using two non-parallel direction vectors in the plane

Parametric Equations of Planes

• Three parametric equations express $x$, $y$, and $z$ each as functions of two parameters $s$ and $t$
• Direction vectors $\mathbf{A}$ and $\mathbf{B}$ span the plane; their cross product gives the normal: $\mathbf{n} = \mathbf{A} \times \mathbf{B}$
• Converting to scalar form requires finding the normal and a point, then substituting into $Ax + By + Cz = D$

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Distance Calculations

Distance formulas combine vector operations with geometric insight. Each formula has a specific structure—know which one applies to each situation.

Distance Between Points, Lines, and Planes

• Point-to-point: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$, the direct extension of the 2D distance formula
• Point-to-plane: $d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$, where the plane is $Ax + By + Cz + D = 0$
• Point-to-line uses projection: find the component of the point-to-line vector that's perpendicular to the direction vector, often via $\frac{|\overrightarrow{PQ} \times \mathbf{v}|}{|\mathbf{v}|}$

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Vector-Valued Functions and Motion

This section bridges vectors with calculus. Vector-valued functions let you describe paths through space and analyze motion using derivatives.

Vector-Valued Functions and Space Curves

• Definition: $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ assigns a position vector to each value of parameter $t$
• Space curves are the paths traced by $\mathbf{r}(t)$; examples include helices, circles in 3D, and parametric representations of intersections
• Derivatives give velocity ($\mathbf{r}'(t)$) and acceleration ($\mathbf{r}''(t)$), connecting vector geometry to physics and kinematics

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Quick Reference Table

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Self-Check Questions

1. What geometric condition does $\mathbf{A} \cdot \mathbf{B} = 0$ indicate, and how does this differ from what $\mathbf{A} \times \mathbf{B} = \mathbf{0}$ tells you?

2. Given two points in 3D space, describe the steps to write both the vector equation and parametric equations for the line through them.

3. Compare and contrast finding the distance from a point to a plane versus the distance from a point to a line—what vector operations does each require?

4. If you need a vector perpendicular to a plane, which product do you use and what inputs do you need? How does this connect to the scalar equation of a plane?

5. Explain how the projection $\text{proj}_{\mathbf{B}}\mathbf{A}$ relates to the dot product, and give one physical situation where this decomposition is useful.