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Mom's Basement University

Introduction to Vectors

What You’ll Learn

Section titled “What You’ll Learn”

If vectors feel like a big jump, don’t fret. This is just an overview. You’ll see vectors again in Calculus and they get a thorough deep dive in Linear Algebra, so think of this as a first look rather than the final word.

In this lesson you’ll learn what vectors are, how to represent them in component form, calculate their magnitude and direction, and perform basic operations like addition, subtraction, and scalar multiplication.

The Concept

Section titled “The Concept”

A vector is a quantity that has both magnitude (how much) and direction (which way). This is different from a scalar, which has only magnitude. Temperature is a scalar (just a number). Velocity is a vector (speed plus direction).

In component form, a vector v is written as an ordered pair of numbers in angle brackets: v = ⟨a, b⟩. The first component is the horizontal part, the second is the vertical part. You can also write it as v = ai + bj, where i and j are the unit vectors along the x and y axes.

A few symbols to know going forward:

  • ⟨a, b⟩ are angle brackets, used specifically for vectors (not the same as parentheses or less-than/greater-than signs)
  • ∣v∣|\mathbf{v}| means the magnitude (length) of vector v, similar to how absolute value gives the “size” of a number
  • Boldface letters like v\mathbf{v} or u\mathbf{u} indicate vectors, while regular letters like k or c are scalars (plain numbers)

The diagram shows a vector v = ⟨3, 4⟩. The orange dashed lines are the horizontal and vertical components (3 right, 4 up). The blue arrow is the vector itself. The magnitude (length) is found using the Pythagorean theorem, and the direction angle is measured from the positive x-axis.

The magnitude (length) of a vector is found using the Pythagorean theorem:

∣v∣=a2+b2|\mathbf{v}| = \sqrt{a^2 + b^2}

The direction angle is the angle the vector makes with the positive x-axis:

θ=tan⁡−1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right)

(Adjust for the correct quadrant, just like with polar coordinates.)

Basic operations work component by component:

  • Scalar multiplication: k⟨a, b⟩ = ⟨ka, kb⟩ (stretches or shrinks the vector)
  • Addition: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩
  • Subtraction: ⟨a, b⟩ - ⟨c, d⟩ = ⟨a - c, b - d⟩

Scalar multiplication changes the length of a vector. Multiplying by 2 doubles the length (green). Multiplying by -1 reverses the direction (red). The direction stays the same for positive scalars and flips for negative ones.

Graphically, you add vectors using the tip-to-tail method: place the tail of the second vector at the tip of the first, then draw the resultant from the origin to the new tip.

Worked Example

Section titled “Worked Example”

Let u = ⟨2, 5⟩ and v = ⟨-3, 4⟩.

Magnitude of u:

∣u∣=22+52=4+25=29≈5.39|\mathbf{u}| = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39

Vector addition (tip-to-tail):

u+v=⟨2+(−3),  5+4⟩=⟨−1,  9⟩\mathbf{u} + \mathbf{v} = \langle 2 + (-3),\; 5 + 4 \rangle = \langle -1,\; 9 \rangle

The diagram shows u (blue) from the origin to (2, 5), then v (orange) from the tip of u to (-1, 9). The dashed green arrow is the resultant u + v from the origin directly to (-1, 9).

Scalar multiplication:

3u=⟨3×2,  3×5⟩=⟨6,  15⟩3\mathbf{u} = \langle 3 \times 2,\; 3 \times 5 \rangle = \langle 6,\; 15 \rangle

This triples the length of u while keeping the same direction.

Direction of u:

θ=tan⁡−1(52)≈68.2°\theta = \tan^{-1}\left(\frac{5}{2}\right) \approx 68.2°

Since both components are positive, u points into the first quadrant.

Real-World Application

Section titled “Real-World Application”

Vectors are essential in:

  • Physics (forces, velocity, acceleration, momentum)
  • Navigation (airplane heading combined with wind gives ground velocity)
  • Engineering (structural forces, fluid flow analysis)
  • Computer graphics (character movement, camera angles, lighting)
  • GPS and mapping (displacement between waypoints)

Example: an airplane flies at 400 mph due east (airspeed vector), but there’s a 50 mph wind blowing from the south (wind vector pointing north). The actual ground velocity is the vector sum of these two, giving a slightly faster speed at a slight northward angle.

Quiz

Section titled “Quiz”
A vector differs from a scalar because it has:
The magnitude of $\langle 6, 8 \rangle$ is:
Vector addition is done by:
Scalar multiplication of a vector means:
Vectors are commonly used to represent: