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.

Definition of Vectors in 3D Space

A vector in 3D is an ordered triplet written as $\langle x, y, z \rangle$, representing a displacement (direction and magnitude) rather than a single point. Its magnitude (or length) is calculated as:

$|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}$

This is just the 3D version of the Pythagorean theorem. Geometrically, you can visualize vectors as arrows in space, which makes operations like addition (tip-to-tail) and scalar multiplication (stretching or shrinking the arrow) intuitive.

A unit vector has magnitude 1 and points in a specific direction. To find the unit vector in the direction of $\mathbf{v}$, divide by its magnitude: $\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$. This comes up constantly when you need direction without caring about length.

Vector Operations (Addition, Subtraction, Scalar Multiplication)

All three operations work component-wise. For $\mathbf{A} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{B} = \langle b_1, b_2, b_3 \rangle$:

• Addition: $\mathbf{A} + \mathbf{B} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle$
• Subtraction: $\mathbf{A} - \mathbf{B} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle$
• Scalar multiplication: $k\mathbf{A} = \langle ka_1, ka_2, ka_3 \rangle$, where a 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. FRQs often ask you to express a new vector as a linear combination, requiring both operations together.

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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

The dot product takes two vectors and returns a scalar (a number, not a vector).

• Algebraic formula: $\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3$
• Geometric formula: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta$

The geometric formula is where the real insight lives. Since the result depends on $\cos\theta$:

• Positive dot product → angle between vectors is acute (less than 90°)
• Zero dot product → vectors are perpendicular (orthogonal)
• Negative dot product → angle between vectors is obtuse (greater than 90°)

To find the angle between two vectors, rearrange: $\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}$. Key properties include commutativity ($\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$) and distributivity over addition.

Cross Product and Its Properties

The cross product takes two vectors and returns a vector perpendicular to both inputs.

• Magnitude: $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin\theta$, which equals the area of the parallelogram formed by the two vectors
• Direction: determined by the right-hand rule (curl fingers from $\mathbf{A}$ toward $\mathbf{B}$; your thumb points in the direction of $\mathbf{A} \times \mathbf{B}$)

To compute $\mathbf{A} \times \mathbf{B}$, use the determinant method:

$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

This expands to $\langle a_2b_3 - a_3b_2,\; a_3b_1 - a_1b_3,\; a_1b_2 - a_2b_1 \rangle$.

The cross product is anti-commutative: $\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}$. Order matters. Also note: if $\mathbf{A} \times \mathbf{B} = \mathbf{0}$, the vectors are parallel (since $\sin\theta = 0$).

Projections of Vectors

The projection of $\mathbf{A}$ onto $\mathbf{B}$ gives you the component of $\mathbf{A}$ that points in the direction of $\mathbf{B}$:

$\text{proj}_{\mathbf{B}}\mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2}\mathbf{B}$

This is a vector. The scalar version (the signed length of the projection) is called the scalar component:

$\text{comp}_{\mathbf{B}}\mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}$

Projection lets you decompose any vector into a parallel part and a perpendicular part relative to another vector. The perpendicular part is just $\mathbf{A} - \text{proj}_{\mathbf{B}}\mathbf{A}$. Physical applications include resolving forces along a direction and calculating work: $W = \mathbf{F} \cdot \mathbf{d}$.

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Equations of Lines and Planes

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

Vector Equations of Lines

A line in 3D is determined by a point and a direction. The vector equation is:

$\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$

Here $\mathbf{r}_0$ is the position vector of a known point on the line, $\mathbf{v}$ is the direction vector, and the parameter $t$ ranges over all real numbers. Each value of $t$ gives a different point on the line.

To find the direction vector between two points $P$ and $Q$, just subtract: $\mathbf{v} = \overrightarrow{PQ} = Q - P$.

Parametric Equations of Lines

The parametric form breaks the vector equation into components:

$x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct$

where $\langle a, b, c \rangle$ is the direction vector and $(x_0, y_0, z_0)$ is a point on the line.

Symmetric equations eliminate $t$ by setting the expressions equal:

$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$

Watch out: if any direction component is zero (say $a = 0$), you can't divide by it. Instead, that coordinate is constant ($x = x_0$), and you write the symmetric equation using only the remaining components.

Converting between vector, parametric, and symmetric forms is a common exam task. Practice going in all directions.

Vector Equations of Planes

A plane is determined by a point and a normal vector (a vector perpendicular to the plane). The point-normal form is:

$\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$

This says: any vector lying in the plane is perpendicular to the normal. Expanding this gives the scalar equation:

$Ax + By + Cz = D$

where $\langle A, B, C \rangle$ is the normal vector and $D = Ax_0 + By_0 + Cz_0$.

Alternatively, the two-parameter form uses two non-parallel direction vectors $\mathbf{u}$ and $\mathbf{w}$ that lie in the plane:

$\mathbf{r}(s, t) = \mathbf{r}_0 + s\mathbf{u} + t\mathbf{w}$

Parametric Equations of Planes

The two-parameter vector equation above expands into three parametric equations (one each for $x$, $y$, $z$) as functions of $s$ and $t$.

The direction vectors $\mathbf{u}$ and $\mathbf{w}$ span the plane. Their cross product gives the normal vector: $\mathbf{n} = \mathbf{u} \times \mathbf{w}$. This is how you convert from parametric form to scalar form:

1. Compute $\mathbf{n} = \mathbf{u} \times \mathbf{w}$ to get $\langle A, B, C \rangle$
2. Pick any known point on the plane (use $\mathbf{r}_0$)
3. Substitute into $Ax + By + Cz = D$ to find $D$

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Distance Calculations

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

Distance Between Points, Lines, and Planes

Point-to-point distance is the direct 3D extension of the Pythagorean theorem:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Point-to-plane distance uses the scalar equation of the plane. If the plane is $Ax + By + Cz + D = 0$ and the point is $(x_0, y_0, z_0)$:

$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$

Note the plane equation must be in the form $Ax + By + Cz + D = 0$ (with $D$ on the left side) for this formula to work directly.

Point-to-line distance requires a cross product. If the line passes through point $P$ with direction $\mathbf{v}$, and you want the distance from point $Q$ to the line:

$d = \frac{|\overrightarrow{PQ} \times \mathbf{v}|}{|\mathbf{v}|}$

The cross product gives the area of the parallelogram formed by $\overrightarrow{PQ}$ and $\mathbf{v}$, and dividing by the base length $|\mathbf{v}|$ gives the height, which is the perpendicular distance.

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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

A vector-valued function assigns a position vector to each value of a parameter $t$:

$\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$

The path traced out by $\mathbf{r}(t)$ as $t$ varies is called a space curve. Common examples include helices ($\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$), circles in 3D, and curves formed by the intersection of two surfaces.

Differentiation connects vector geometry to motion:

• $\mathbf{r}'(t)$ gives the velocity vector, which is tangent to the curve at each point
• $\mathbf{r}''(t)$ gives the acceleration vector
• Differentiation is done component-wise: $\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$

A parametric line ($\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$) is the simplest vector-valued function since its components are linear in $t$. Curves with nonlinear components require more careful analysis of tangent vectors and arc length.

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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.