Vectors

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Represent vectors graphically and in component form.
  • Add, subtract, and scale vectors using components.
  • Compute magnitude and direction.
  • Apply vectors to displacement, velocity, and force problems.

Key Ideas

A vector has:

  • magnitude (length)
  • direction

Common representations:

  • Arrow in the plane
  • Component form:
    \[ \langle a,\, b \rangle \]
  • As difference of two points:
    \[ \overrightarrow{AB} = \langle x_B - x_A,\; y_B - y_A \rangle \]

Magnitude of vector \(\langle a, b \rangle\): \[ \|\langle a, b \rangle\| = \sqrt{a^2 + b^2} \]

Direction angle \(\theta\) (from positive x-axis): \[ \tan \theta = \frac{b}{a} \]

Operations:

  • Addition:
    \[ \langle a,b \rangle + \langle c,d \rangle = \langle a+c,\, b+d \rangle \]

  • Subtraction:
    \[ \langle a,b \rangle - \langle c,d \rangle = \langle a-c,\, b-d \rangle \]

  • Scalar multiplication:
    \[ k\langle a,b \rangle = \langle ka,\, kb \rangle \]

Common Problem Types

Finding a Vector From Two Points

Use the difference of coordinates.

Example:
\(A(1,2)\), \(B(6,5)\)
\[ \overrightarrow{AB} = \langle 6-1,\; 5-2 \rangle = \langle 5,3 \rangle \]

Magnitude of a Vector

Use the distance formula.

Example:
\[ \|\langle 4, -3 \rangle\| = 5 \]

Adding or Subtracting Vectors

Add or subtract components.

Example:
\[ \langle 2,5 \rangle + \langle -1,4 \rangle = \langle 1,9 \rangle \]

Scalar Multiples

Multiply each component.

Example:
\[ 3\langle 2,-1 \rangle = \langle 6,-3 \rangle \]

Displacement / Velocity Applications

Vectors represent movement in the plane.

Example:
Walk \(\langle 3,4 \rangle\) blocks, then \(\langle -1,2 \rangle\):
Total \(= \langle 2,6 \rangle\).

Strategies

  • Draw arrows to visualize direction.
  • Combine components carefully — avoid mixing x and y.
  • Scalar multiplication stretches or shrinks magnitude.
  • Check quadrant when finding direction angle.
  • Magnitude is always nonnegative.

Worked Examples

Example 1 — Find magnitude and direction

Given \(\vec{v} = \langle 6, 8 \rangle\):

Magnitude: \[ \sqrt{6^2 + 8^2} = 10 \]

Direction: \[ \tan \theta = \frac{8}{6} \Rightarrow \theta \approx 53.13^\circ \]

Example 2 — Add vectors

Compute: \[ \langle 4, -1 \rangle + \langle -2, 6 \rangle \]

Solution: \[ \langle 2, 5 \rangle \]

Example 3 — Vector between points

Find \(\overrightarrow{PQ}\) for \(P(-3,1)\) and \(Q(5,4)\):

\[ \langle 5 - (-3),\, 4 - 1 \rangle = \langle 8, 3 \rangle \]

Example 4 — Displacement

A plane travels east \(300\) miles and north \(400\) miles.

Total displacement vector: \[ \langle 300, 400 \rangle \]

Magnitude: \[ \sqrt{300^2 + 400^2} = 500 \text{ miles} \]

Direction: \[ \tan \theta = \frac{400}{300} \]

WarningCommon Mistakes
  • Mixing up direction angle with slope.
  • Forgetting to square components when finding magnitude.
  • Switching the order in \(\overrightarrow{AB}\) (should be \(B-A\)).
  • Treating vectors as points instead of directed segments.
  • Sign errors when adding/subtracting.

Practice Problems

  1. Find the magnitude: \(\langle -3, 4 \rangle\).
  2. Add: \(\langle 1,2 \rangle + \langle 5,-7 \rangle\).
  3. Find \(\overrightarrow{AB}\) for \(A(2,-1)\) and \(B(7,5)\).
  4. Multiply: \(-2\langle 3, -4 \rangle\).
  5. A person walks \(\langle 6,2 \rangle\) miles, then \(\langle -4,5 \rangle\). Find net displacement.

1.
\[ \sqrt{(-3)^2 + 4^2} = 5 \]

2.
\[ \langle 6, -5 \rangle \]

3.
\[ \langle 7-2,\; 5-(-1) \rangle = \langle 5, 6 \rangle \]

4.
\[ \langle -6, 8 \rangle \]

5.
Net: \(\langle 2, 7 \rangle\)

Summary

  • Vectors have magnitude and direction.
  • Add components; scale components; magnitude uses the Pythagorean formula.
  • Vectors model displacement, velocity, and physical forces.
  • Draw a sketch for displacement problems.
  • Use \(B - A\) for \(\overrightarrow{AB}\).
  • Direction angle: check the quadrant.
  • Magnitude is always nonnegative.