Chapter 14

Medium

Vectors in 2D/3D

00 · Symbol Glossary

$\vec{v}$v arrow — directed quantity

A quantity with both a magnitude (size) and a direction — drawn as an arrow. The arrow notation $\vec{v}$ distinguishes a directed quantity from a plain number like $x$ or $t$ , which carries size only.

$\langle a, b\rangle$component form (2D)

A directed quantity written as an ordered pair: $a$ is how far it points in the horizontal direction, $b$ how far in the vertical direction. Angle brackets (not the round parentheses used for a point, Chapter 13) mark this as a displacement — "go $a$ across, $b$ up" — not a location.

$\langle a,b,c\rangle$component form (3D)

The same idea extended to three-dimensional space: a third component $c$ for displacement along a third, mutually perpendicular direction.

$\lVert\vec{v}\rVert$magnitude of v — length

The length of the arrow — always a non-negative number, computed by extending the Pythagorean theorem to however many components $\vec{v}$ has.

$\hat{\imath},\ \hat{\jmath},\ \hat{k}$i-hat, j-hat, k-hat — standard unit vectors

Fixed reference arrows of length $1$ , pointing along the horizontal, vertical, and (in 3D) depth directions respectively. Any directed quantity can be rebuilt from scaled copies of these three: $\langle a,b,c\rangle = a\hat{\imath}+b\hat{\jmath}+c\hat{k}$ .

$\vec{a}\cdot\vec{b}$dot product

An operation combining two directed quantities into a single plain number, measuring how aligned their directions are.

$\vec{a}\times\vec{b}$cross product

An operation (3D only) combining two directed quantities into a new directed quantity, perpendicular to both, whose length measures the area of the parallelogram they frame.

$\vec{r}(t)$r of t — position as a directed quantity

A directed quantity from the origin to a moving point, built from the parametric coordinate functions of Chapter 13: $\vec{r}(t) = \langle x(t), y(t)\rangle$ . Differentiating it in Section 05 recovers the direction and rate of motion along the curve.

01 · Directed Quantities

Some quantities are fully described by a single number: temperature, mass, the value $f(3)$ . Others need more — "the wind is blowing" is incomplete without also saying which way and how hard. A quantity with both a size and a direction is called a directed quantity, or vector, and is drawn as an arrow.

Definition — Vector (Directed Quantity)

A vector $\vec{v}$ is a quantity possessing both a magnitude $\lVert\vec{v}\rVert\geq0$ and a direction. Two arrows represent the same vector if they have the same length and point the same way — regardless of where they are drawn; only the displacement they encode matters, not their starting point.

Definition — Component Form

Placing a vector's tail at the origin, its tip lands at some point $(a,b)$ (in the plane) or $(a,b,c)$ (in space). The component form records exactly that displacement:

\vec{v} = \langle a, b\rangle \quad \text{(2D)} \qquad \vec{v} = \langle a, b, c\rangle \quad \text{(3D)}

A vector is a displacement, not a location

$\langle 3, 4\rangle$ drawn from the origin ends at the point $(3,4)$ — but the same arrow, slid to start at $(1,1)$ , ends at $(4,5)$ instead, and is still the same vector $\langle3,4\rangle$ , because it represents the same "go 3 across, 4 up" instruction. A point is a fixed location; a vector is an instruction for how to move.

02 · Combining Vectors: Sum and Scalar Multiple

Two operations combine directed quantities: adding two of them together, and scaling a single one by a plain number.

Definition — Vector Sum and Scalar Multiple

\langle a_1,b_1\rangle + \langle a_2,b_2\rangle = \langle a_1+a_2,\ b_1+b_2\rangle \qquad c\langle a,b\rangle = \langle ca,\ cb\rangle

(with an analogous third component in 3D). $c$ is a plain number — no direction of its own — that stretches, shrinks, or reverses $\vec{v}$ .

Example — Adding displacements

Walking along $\vec{u}=\langle2,5\rangle$ (2 east, 5 north, in some fixed units) and then along $\vec{v}=\langle4,-1\rangle$ (4 east, 1 south) lands at the combined displacement $\vec{u}+\vec{v} = \langle2+4,\ 5+(-1)\rangle = \langle6,4\rangle$ — geometrically, placing the tail of the second arrow at the tip of the first, the combined arrow runs from the very start straight to the very end.

Step-by-step — Compute $\lVert\vec{v}\rVert$ for $\vec{v}=\langle-3,0,4\rangle$

Square each component:

(-3)^2=9

0^2=0

4^2=16

Sum the squares:

9+0+16=25

Take the square root (extending the Pythagorean theorem to three components):

\lVert\vec{v}\rVert = \sqrt{25}=5

03 · The Dot Product

Definition — Dot Product

\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+a_3b_3 \qquad \text{equivalently} \qquad \vec{a}\cdot\vec{b} = \lVert\vec{a}\rVert\,\lVert\vec{b}\rVert\cos\theta

where $\theta$ is the angle between the two directed quantities when their tails are placed together. The result is a plain number, not a new directed quantity.

Example — Sign of the dot product reveals alignment

$\vec{a}=\langle1,0\rangle$ , $\vec{b}=\langle0,1\rangle$ : $\vec{a}\cdot\vec{b} = 1(0)+0(1)=0$ . A dot product of $0$ means $\theta=90°$ (since $\cos90°=0$ ) — the two directions are perpendicular.

04 · The Cross Product (3D)

The dot product collapses two directed quantities into a plain number. The cross product instead produces a new directed quantity — one perpendicular to both original ones, defined only in three dimensions.

Definition — Cross Product

For $\vec{a}=\langle a_1,a_2,a_3\rangle$ , $\vec{b}=\langle b_1,b_2,b_3\rangle$ :

\vec{a}\times\vec{b} = \langle\, a_2b_3-a_3b_2,\ \ a_3b_1-a_1b_3,\ \ a_1b_2-a_2b_1\,\rangle

Each component pattern skips the matching index: the first component uses only the $2$ nd and $3$ rd components of $\vec a,\vec b$ ; the second uses only the $3$ rd and $1$ st; the third uses only the $1$ st and $2$ nd.

The result $\vec{a}\times\vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ , with $\lVert\vec{a}\times\vec{b}\rVert = \lVert\vec{a}\rVert\lVert\vec{b}\rVert\sin\theta$ — the area of the parallelogram framed by the two directed quantities.

Step-by-step — Compute $\vec{a}\times\vec{b}$ for $\vec{a}=\langle1,2,0\rangle$, $\vec{b}=\langle0,3,1\rangle$

First component — skip index 1, use indices 2,3:

a_2b_3-a_3b_2 = (2)(1)-(0)(3) = 2-0=2

Second component — skip index 2, use indices 3,1:

a_3b_1-a_1b_3 = (0)(0)-(1)(1) = 0-1=-1

Third component — skip index 3, use indices 1,2:

a_1b_2-a_2b_1 = (1)(3)-(2)(0)=3-0=3

Assemble:

\vec{a}\times\vec{b} = \langle2,-1,3\rangle

Verify perpendicularity using the dot product (Section 03):

\vec{a}\cdot(\vec{a}\times\vec{b}) = 1(2)+2(-1)+0(3) = 2-2+0=0

✓;

\vec{b}\cdot(\vec{a}\times\vec{b}) = 0(2)+3(-1)+1(3) = 0-3+3=0

✓. Both dot products are

0

, confirming

\vec{a}\times\vec{b}

is perpendicular to both original directed quantities.

❌ Assuming the cross product commutes like the dot product

For $\vec{a}=\langle1,0,0\rangle$ , $\vec{b}=\langle0,1,0\rangle$ : $\vec{a}\times\vec{b} = \langle(0)(0)-(0)(1),\ (0)(0)-(1)(0),\ (1)(1)-(0)(0)\rangle = \langle0,0,1\rangle$ .

Now swap the order: $\vec{b}\times\vec{a} = \langle(1)(0)-(0)(0),\ (0)(0)-(0)(0),\ (0)(0)-(1)(1)\rangle = \langle0,0,-1\rangle$ .

Why it breaks: unlike the dot product ( $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$ , since ordinary multiplication of the matching components doesn't care about order), the cross product's component pattern flips sign when the two directed quantities are swapped — each term is a difference, and swapping $\vec a,\vec b$ swaps the minuend and subtrahend in every component.

Consequence: $\vec{a}\times\vec{b} = \langle0,0,1\rangle \neq \langle0,0,-1\rangle = \vec{b}\times\vec{a}$ — in fact $\vec{b}\times\vec{a} = -(\vec{a}\times\vec{b})$ always. Order matters for the cross product; it never commutes except in the trivial case where the result is the zero vector.

05 · Vector-Valued Functions of a Parameter

Chapter 13's parametric curve $x(t), y(t)$ can be repackaged as a single directed quantity that changes with $t$ — bundling both coordinate functions into one object.

Definition — Vector-Valued Function and Its Derivative

\vec{r}(t) = \langle x(t),\ y(t)\rangle \qquad \vec{r}\,'(t) = \langle x'(t),\ y'(t)\rangle

$\vec{r}(t)$ traces the same curve as Chapter 13's $x(t),y(t)$ , viewed as a directed quantity from the origin to the moving point. Each component of $\vec{r}\,'(t)$ is differentiated exactly as in Chapter 02 — componentwise.

Geometric meaning of the derivative

$\vec{r}\,'(t)$ points in the direction the curve is currently heading at parameter value $t$ — tangent to the path, not perpendicular to it. If $t$ is interpreted as time, $\vec{r}\,'(t)$ is the instantaneous velocity: its direction is which way motion is heading, and its magnitude $\lVert\vec{r}\,'(t)\rVert$ is the speed — a single non-negative number, matching Chapter 02's ordinary derivative-as-rate-of-change idea, just carrying a direction alongside the rate now.

Step-by-step — Find the velocity and speed of $\vec{r}(t) = \langle t^2, t^3-3t\rangle$ at $t=2$

Differentiate each component (Chapter 02, power rule):

x'(t)=2t

;

y'(t)=3t^2-3

Assemble the velocity vector:

\vec{r}\,'(t) = \langle2t,\ 3t^2-3\rangle

Substitute $t=2$ :

\vec{r}\,'(2) = \langle4,\ 9\rangle

(matching Chapter 13, Section 01's slope example on this same curve:

\dfrac{dy}{dx}=\dfrac{9}{4}

, exactly the ratio of these two components).

Compute the speed — the magnitude of the velocity vector (Section 02's formula):

\lVert\vec{r}\,'(2)\rVert = \sqrt{4^2+9^2} = \sqrt{16+81}=\sqrt{97}\approx9.85

❌ Confusing the magnitude of a sum with the sum of magnitudes

For $\vec{u}=\langle3,0\rangle$ and $\vec{v}=\langle0,4\rangle$ : a tempting shortcut is $\lVert\vec u+\vec v\rVert \overset{?}{=} \lVert\vec u\rVert+\lVert\vec v\rVert = 3+4=7$ .

Why it breaks: $\vec u+\vec v = \langle3,4\rangle$ , so the actual magnitude is $\lVert\vec u+\vec v\rVert = \sqrt{3^2+4^2}=\sqrt{25}=5$ , not $7$ . Adding lengths ignores that the two directed quantities point in different directions — combining magnitudes directly only works when two vectors point the exact same way.

Consequence: $5 < 7$ here — in general $\lVert\vec u+\vec v\rVert \leq \lVert\vec u\rVert+\lVert\vec v\rVert$ (equality only when $\vec u,\vec v$ point in the same direction), so magnitudes never simply add unless direction is accounted for.

06 · Exercises

EXERCISE 14.1

Add corresponding components; scale every component by the given constant.

$\vec{u}=\langle2,-3,1\rangle$ , $\vec{v}=\langle-1,4,2\rangle$ . $\vec u+\vec v = \langle2+(-1),\,-3+4,\,1+2\rangle = \langle1,1,3\rangle$ . $3\vec u = \langle6,-9,3\rangle$ .

Compute $\vec u+\vec v$ and $3\vec u$ for $\vec u=\langle2,-3,1\rangle$ , $\vec v=\langle-1,4,2\rangle$ .

EXERCISE 14.2

Multiply matching components and sum; the sign of the result reveals whether the angle between the vectors is acute, right, or obtuse.

$\vec a=\langle2,-1,3\rangle$ , $\vec b=\langle1,4,1\rangle$ . $\vec a\cdot\vec b = 2(1)+(-1)(4)+3(1) = 2-4+3=1$ . Positive, so the angle between them is acute (less than $90°$ ).

Compute $\vec a\cdot\vec b$ for $\vec a=\langle2,-1,3\rangle$ , $\vec b=\langle1,4,1\rangle$ , and state whether the angle between them is acute, right, or obtuse.

EXERCISE 14.3

Apply the component-skipping pattern from Section 04 for each of the three output components.

$\vec a=\langle3,0,-1\rangle$ , $\vec b=\langle1,2,2\rangle$ . First: $a_2b_3-a_3b_2 = 0(2)-(-1)(2)=0+2=2$ . Second: $a_3b_1-a_1b_3=(-1)(1)-3(2)=-1-6=-7$ . Third: $a_1b_2-a_2b_1=3(2)-0(1)=6-0=6$ . $\vec a\times\vec b = \langle2,-7,6\rangle$ .

Compute $\vec a\times\vec b$ for $\vec a=\langle3,0,-1\rangle$ , $\vec b=\langle1,2,2\rangle$ .

EXERCISE 14.4

Differentiate each component of the position vector separately, then compute the magnitude of the resulting velocity vector.

$\vec r(t) = \langle\cos t,\ \sin t\rangle$ . $\vec r\,'(t) = \langle-\sin t,\ \cos t\rangle$ (Chapter 02, Section 09). At $t=\dfrac\pi2$ : $\vec r\,'\left(\frac\pi2\right) = \langle-1, 0\rangle$ . Speed: $\lVert\vec r\,'(\pi/2)\rVert = \sqrt{(-1)^2+0^2}=1$ — constant speed $1$ at every $t$ , in fact, since $\lVert\vec r\,'(t)\rVert=\sqrt{\sin^2t+\cos^2t}=1$ always.

Find the velocity vector and speed of $\vec r(t) = \langle\cos t,\ \sin t\rangle$ at $t=\dfrac{\pi}{2}$ .

EXERCISE 14.5

Recall which operation from this chapter changes sign when its two inputs are swapped, and by how much.

The cross product anticommutes: $\vec a\times\vec b = -(\vec b\times\vec a)$ (Section 04's FailBlock). The dot product commutes: $\vec a\cdot\vec b=\vec b\cdot\vec a$ , since it is built from ordinary multiplication of matching components, and ordinary multiplication does not care about order.

Explain, using the definitions from Sections 03–04, why the dot product is symmetric in its two inputs ( $\vec a\cdot\vec b=\vec b\cdot\vec a$ ) while the cross product is not.

EXERCISE 14.6

A price and a hedge ratio each moving in a specific "direction" per unit of a driver quantity is naturally modeled as a directed quantity — combine two such quantities by adding components.

Two trading signals produce per-day directional adjustments $\vec s_1 = \langle 0.02, -0.01\rangle$ and $\vec s_2 = \langle -0.03, 0.04\rangle$ (units: fractional position change in two assets). Combined daily adjustment: $\vec s_1+\vec s_2 = \langle 0.02-0.03,\ -0.01+0.04\rangle = \langle-0.01,\ 0.03\rangle$ . Magnitude of the combined adjustment: $\lVert\vec s_1+\vec s_2\rVert = \sqrt{(-0.01)^2+(0.03)^2}=\sqrt{0.0001+0.0009}=\sqrt{0.001}\approx0.0316$ .

Two trading signals produce daily directional position adjustments $\vec s_1=\langle0.02,-0.01\rangle$ and $\vec s_2=\langle-0.03,0.04\rangle$ across two assets. Find the combined daily adjustment and its magnitude.

07 · Chapter Summary

Concept	Formula / Rule
Vector	Directed quantity: magnitude + direction, drawn as an arrow
Component form	$\langle a,b\rangle$ (2D) or $\langle a,b,c\rangle$ (3D) — a displacement, not a location
Magnitude	$\lVert\vec v\rVert = \sqrt{a^2+b^2(+c^2)}$
Sum / scalar multiple	Componentwise addition; scale every component by $c$
Dot product	$\vec a\cdot\vec b = \sum a_ib_i = \lVert\vec a\rVert\lVert\vec b\rVert\cos\theta$ ; commutes
Cross product (3D)	$\vec a\times\vec b$ , perpendicular to both; anticommutes: $\vec a\times\vec b=-\vec b\times\vec a$
Vector-valued function	$\vec r(t)=\langle x(t),y(t)\rangle$ ; $\vec r\,'(t)$ = velocity, $\lVert\vec r\,'(t)\rVert$ = speed

Next: Chapter 15 — Partial Derivatives, which extends the derivative to functions of more than one variable, using directed quantities to describe the direction of steepest change.