10.8 Vectors

Vector Fundamentals

Vectors let you represent quantities that have both a size and a direction. Unlike regular numbers (scalars), which only tell you "how much," vectors also tell you "which way." This makes them essential for describing things like forces, velocities, and displacements.

Vectors as geometric objects

A vector is drawn as a directed line segment: its length represents the vector's magnitude (how big it is), and the arrow indicates its direction (where it points).

• Vectors are denoted using boldface letters ($\mathbf{v}$) or letters with arrows ($\vec{v}$)
• The magnitude of $\vec{v}$ is written as $|\vec{v}|$
• Two vectors are equal if they have the same magnitude and direction, even if they're drawn at different positions in space

Vector magnitude and direction

For a vector $\vec{v} = (x, y)$ in the coordinate plane, you can find its magnitude using the Pythagorean theorem:

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

For example, the vector $\vec{v} = (3, 4)$ has magnitude $|\vec{v}| = \sqrt{9 + 16} = 5$.

The direction of a vector is the angle it makes with the positive x-axis. You find it with the arctangent function:

$\theta = \tan^{-1}\left(\frac{y}{x}\right)$

One thing to watch out for: $\tan^{-1}$ only gives you angles in Quadrants I and IV. If your vector points into Quadrant II or III (meaning $x < 0$), you need to add $180°$ to the result to get the correct angle.

Basic vector operations

Vector addition: Add corresponding components.

$\vec{u} + \vec{v} = (u_1 + v_1,\; u_2 + v_2)$

For example, $(2, 3) + (1, -5) = (3, -2)$. Geometrically, you can visualize this by placing the tail of $\vec{v}$ at the head of $\vec{u}$ (the head-to-tail method). The resultant vector goes from the tail of $\vec{u}$ to the head of $\vec{v}$.

Scalar multiplication: Multiply each component by the scalar.

$c\vec{v} = (cx,\; cy)$

This scales the magnitude by $|c|$. If $c$ is positive, the direction stays the same. If $c$ is negative, the vector flips to point in the opposite direction.

Vector Representation and Products

Vectors in component form

The standard unit vectors in the plane are:

• $\hat{i} = (1, 0)$, pointing in the positive x-direction
• $\hat{j} = (0, 1)$, pointing in the positive y-direction

Any vector $\vec{v} = (x, y)$ can be written as a linear combination of these unit vectors:

$\vec{v} = x\hat{i} + y\hat{j}$

So the vector $(3, -2)$ becomes $3\hat{i} - 2\hat{j}$. This notation makes addition and scalar multiplication straightforward since you just work with the $\hat{i}$ and $\hat{j}$ components separately.

Dot product of vectors

The dot product of $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$ is:

$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2$

Notice the result is a scalar (a number), not a vector.

The dot product is also related to the angle $\theta$ between the two vectors:

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

This gives you two useful applications:

• Finding the angle between vectors: Rearrange to get $\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|\;|\vec{v}|}$
• Testing for perpendicularity: Two vectors are orthogonal (perpendicular) if and only if $\vec{u} \cdot \vec{v} = 0$, because $\cos(90°) = 0$

Unit vectors in direction

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

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

For example, the unit vector in the direction of $(3, 4)$ is $\frac{1}{5}(3, 4) = (0.6,\; 0.8)$. Unit vectors are useful when you care about direction only, without any particular magnitude attached.

Applications of Vectors

Vector applications in physics

Vectors represent physical quantities that have both magnitude and direction: displacement, velocity, acceleration, and force are the most common examples.

A typical application is finding the resultant force on an object when multiple forces act on it:

1. Express each force as a vector in component form
2. Add all the force vectors component-wise to get the resultant vector
3. Calculate the magnitude of the resultant using $|\vec{v}| = \sqrt{x^2 + y^2}$
4. Find the direction using $\theta = \tan^{-1}\left(\frac{y}{x}\right)$, adjusting for the correct quadrant

The dot product also shows up in physics through the concept of work: the work done by a force $\vec{F}$ over a displacement $\vec{d}$ is $W = \vec{F} \cdot \vec{d}$. This captures the idea that only the component of force in the direction of motion does work.