Vectors
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 2.1: “Vectors in the Plane” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/2-1-vectors-in-the-plane |
| Textbook used in class | Stewart, Calculus, Section 12.2: “Vectors” (Examples 1, 2, 3) |
Opening Scenario
A pilot reports “wind at 30 knots from the northwest.” That phrase contains two pieces of information: a magnitude (30 knots) and a direction (from the northwest). A scalar like temperature carries only magnitude. A vector carries both. Vectors model forces, velocities, displacements, and any other quantity where direction matters.
Quick Reference
Vector: a quantity with both magnitude and direction. Denoted $\mathbf{a}$, $\vec{a}$, or in component form $\langle a_1, a_2, a_3\rangle$.
Component form. The vector from $A = (x_1, y_1, z_1)$ to $B = (x_2, y_2, z_2)$ is \[ \overrightarrow{AB} = \langle x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1\rangle. \]
Magnitude (length): $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.
Zero vector: $\mathbf{0} = \langle 0, 0, 0\rangle$, with magnitude $0$ and no defined direction.
Equality: $\mathbf{a} = \mathbf{b}$ if and only if $a_1 = b_1$, $a_2 = b_2$, $a_3 = b_3$.
Key Concepts
1. Vectors as Displacements
A vector $\mathbf{v}$ represents a displacement from one point to another. Two arrows with the same length and direction represent the same vector, regardless of where they start. The vector is a free vector: it can be placed anywhere.
The position vector of a point $P = (a_1, a_2, a_3)$ is $\overrightarrow{OP} = \langle a_1, a_2, a_3\rangle$, starting at the origin.
2. Finding Component Form
Example 1. Find the vector represented by the directed line segment from $A = (3, -2, 1)$ to $B = (1, 4, -3)$. (Stewart 12.2, Example 1.)
\[ \overrightarrow{AB} = \langle 1 - 3,\; 4 - (-2),\; -3 - 1\rangle = \langle -2, 6, -4\rangle. \]
Magnitude: $|\overrightarrow{AB}| = \sqrt{4 + 36 + 16} = \sqrt{56} = 2\sqrt{14}$.
3. Standard Basis Vectors
The three standard basis vectors are \[ \mathbf{i} = \langle 1, 0, 0\rangle, \quad \mathbf{j} = \langle 0, 1, 0\rangle, \quad \mathbf{k} = \langle 0, 0, 1\rangle. \] Any vector $\mathbf{a} = \langle a_1, a_2, a_3\rangle$ can be written as $a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$.
treating a vector and a point as the same object. The point $P = (2, 3, 1)$ and the vector $\mathbf{v} = \langle 2, 3, 1\rangle$ use the same three numbers but mean different things. A point is a location; a vector is a displacement. Mixing notation (writing a vector in parentheses, or a point in angle brackets) causes confusion in dot products and cross products.
Common Errors Summary
| Error | Correction |
|---|---|
| Subtracting in the wrong order for $\overrightarrow{AB}$ | $\overrightarrow{AB} = B - A$ (terminal minus initial) |
| Forgetting to take the square root for magnitude | $|\mathbf{a}|$ is the square root of the sum of squares |
Common Misconceptions
a vector and a point with the same coordinates are the same mathematical object.
This is the action-view-of-function error applied to vector notation. The point $P = (2, 3, 1)$ is a location in space; the vector $\mathbf{v} = \langle 2, 3, 1\rangle$ is a displacement. A point cannot be scaled or added to another point component-wise; a vector can. Mixing the two notations causes errors when computing dot products, cross products, and vector equations of lines and planes, all of which require vectors, not points.
Leveled Practice
Problem 1. Find $\overrightarrow{PQ}$ and its magnitude for $P = (0, -1, 2)$ and $Q = (3, 1, 4)$.
Show answer
$\overrightarrow{PQ} = \langle 3, 2, 2\rangle$. $|\overrightarrow{PQ}| = \sqrt{9+4+4} = \sqrt{17}$.