8.8 Vectors

Vector Fundamentals

Vectors describe quantities that have both a size (magnitude) and a direction. Unlike regular numbers (scalars) that only tell you "how much," vectors tell you "how much and which way." This makes them essential for describing things like displacement, velocity, and force.

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Geometric and algebraic vector interpretation

Geometrically, a vector is a directed line segment: it has an initial point (tail) and a terminal point (tip), and the arrow shows which direction it points. You can slide a vector anywhere in the plane without changing it, as long as you keep the same length and direction.

Algebraically, vectors are represented in two common ways:

• Component form: $\vec{v} = \langle a, b \rangle$, where $a$ is the horizontal component and $b$ is the vertical component
• Unit vector notation: $\vec{v} = a\hat{i} + b\hat{j}$, where $\hat{i}$ is the unit vector in the x-direction and $\hat{j}$ is the unit vector in the y-direction

Two vectors are equal if and only if they have the same magnitude and direction. Their starting points don't matter. A vector from $(0,0)$ to $(3,4)$ is the same vector as one from $(1,1)$ to $(4,5)$.

Vector magnitude and direction

The magnitude (or length) of a vector $\vec{v} = \langle a, b \rangle$ comes from the Pythagorean theorem:

$|\vec{v}| = \sqrt{a^2 + b^2}$

For example, if $\vec{v} = \langle 3, 4 \rangle$, then $|\vec{v}| = \sqrt{9 + 16} = 5$.

A vector's direction angle $\theta$ is the angle it makes with the positive x-axis. You find it using:

$\tan \theta = \frac{b}{a}$

Be careful here: $\tan^{-1}\left(\frac{b}{a}\right)$ only gives you the reference angle. You need to check which quadrant the vector points into and adjust $\theta$ accordingly. If $a < 0$, add $180°$ to the calculator output. If $a > 0$ and $b < 0$, add $360°$.

A unit vector has magnitude 1. To find the unit vector in the direction of $\vec{v}$, divide by its magnitude:

$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$

Vector operations

Addition: Add corresponding components.

$\vec{u} + \vec{v} = \langle a_1 + a_2,\; b_1 + b_2 \rangle$

Geometrically, this is the tip-to-tail method: place the tail of $\vec{v}$ at the tip of $\vec{u}$, and the resultant vector goes from the tail of $\vec{u}$ to the tip of $\vec{v}$. Equivalently, the parallelogram law says the resultant is the diagonal of the parallelogram formed by the two vectors.

Subtraction: Subtract corresponding components.

$\vec{u} - \vec{v} = \langle a_1 - a_2,\; b_1 - b_2 \rangle$

Think of this as adding $\vec{u}$ and $-\vec{v}$ (the vector pointing the opposite direction from $\vec{v}$).

Scalar multiplication: Multiply each component by the scalar.

$c\vec{v} = \langle ca,\; cb \rangle$

If $c > 0$, the vector stretches or shrinks but keeps its direction. If $c < 0$, the vector also reverses direction.

Vector Representations and Operations

Operations in i and j notation

All the same operations work in unit vector notation. You just combine the $\hat{i}$ and $\hat{j}$ terms separately:

• Addition: $(a_1\hat{i} + b_1\hat{j}) + (a_2\hat{i} + b_2\hat{j}) = (a_1 + a_2)\hat{i} + (b_1 + b_2)\hat{j}$
• Subtraction: $(a_1\hat{i} + b_1\hat{j}) - (a_2\hat{i} + b_2\hat{j}) = (a_1 - a_2)\hat{i} + (b_1 - b_2)\hat{j}$
• Scalar multiplication: $c(a\hat{i} + b\hat{j}) = (ca)\hat{i} + (cb)\hat{j}$

This notation is just a different way of writing the same component form, so use whichever your course prefers.

Dot product of vectors

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

$\vec{u} \cdot \vec{v} = a_1a_2 + b_1b_2$

For example, $\langle 2, 3 \rangle \cdot \langle 4, -1 \rangle = (2)(4) + (3)(-1) = 8 - 3 = 5$.

The dot product also connects to the angle between two vectors through this formula:

$\vec{u} \cdot \vec{v} = |\vec{u}|\;|\vec{v}|\cos \theta$

Solving for the angle gives you:

$\cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|\;|\vec{v}|}$

Three useful results come from the dot product:

• Orthogonality test: Two vectors are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$
• Angle between vectors: Use the formula above and take $\cos^{-1}$
• Vector projection of $\vec{u}$ onto $\vec{v}$:

$\text{proj}_{\vec{v}}\;\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\;\vec{v}$

This tells you how much of $\vec{u}$ points in the direction of $\vec{v}$.

Advanced vector operations

These topics go slightly beyond standard 2D vectors but are worth knowing at the honors level:

• Linear combination: Any vector can be written as a sum of scalar multiples of other vectors. For instance, $\langle 5, 7 \rangle = 5\hat{i} + 7\hat{j}$ is a linear combination of the standard unit vectors.
• Cross product: This applies to 3D vectors and produces a new vector that is perpendicular to both inputs. You won't typically use it in pre-calc, but it shows up in physics and multivariable calculus.

Vector applications in real-world problems

Vectors model any quantity that has both magnitude and direction. Common applications include:

• Navigation: A pilot flying at 250 mph on a heading of $30°$ with a 40 mph crosswind can find the actual ground velocity by adding the plane's velocity vector and the wind vector.
• Force and equilibrium: When multiple forces act on an object, you add the force vectors. If the resultant is $\langle 0, 0 \rangle$, the object is in equilibrium.
• Work: The work done by a force $\vec{F}$ over a displacement $\vec{d}$ is $W = \vec{F} \cdot \vec{d}$, which uses the dot product.

A general approach for these problems:

1. Identify the vector quantities and draw a diagram
2. Write each vector in component form (use $\langle |\vec{v}|\cos\theta,\; |\vec{v}|\sin\theta \rangle$ to convert from magnitude and angle)
3. Perform the required operations (addition, dot product, etc.)
4. Interpret the result: find the magnitude and/or direction angle of the resultant