MATH 130 · Module 13

Vectors

Concepts, worked examples, student practice, and synthesis

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    represent vectors in component form

  2. 2

    perform vector operations and interpret scalar multiplication

  3. 3

    find vector magnitude and direction angle

Activate · 0.5 minSource: 1

Predict: represent vectors in component form

Predict first

What information does speed omit that velocity includes?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

represent vectors in component form

Concept
  • A vector has magnitude and direction.
  • $\vec v=\langle x_2-x_1,y_2-y_1\rangle$.
  • From magnitude and direction: $\langle |v|\cos\theta,|v|\sin\theta\rangle$.

Explain · 2.5 minSource: 2

Model the Skill: represent vectors in component form

Annotated diagram
  • Find the vector from $A=(-2,5)$ to $B=(3,-1)$.
  • Terminal minus initial gives $\langle3-(-2),-1-5\rangle$.
  • $\vec v=\langle5,-6\rangle$.
  • The corrected diagram moves 5 right and 6 down.
Vector from A negative two comma five to B three comma negative one with components five and negative six
Vector from A negative two comma five to B three comma negative one with components five and negative six

Model · 3.0 minSource: 3

You Try: represent vectors in component form

Student practice

Find the component form from $P=(-4,-2)$ to $Q=(2,1)$.

Practice · 3.0 minSource: 4

Check and Diagnose: represent vectors in component form

Error analysis
  • Use terminal minus initial.
  • Component form describes displacement, not location.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 5

Pacing checkpoint

Make It Stick: represent vectors in component form

Explain both ways to obtain vector components.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 6

Predict: perform vector operations and interpret scalar multiplication

Predict first

What changes when a vector is multiplied by a negative scalar?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

perform vector operations and interpret scalar multiplication

Concept
  • Add and subtract component-wise.
  • $k\langle a,b\rangle=\langle ka,kb\rangle$.
  • A negative scalar reverses direction.

Explain · 2.5 minSource: 7

Model the Skill: perform vector operations and interpret scalar multiplication

Annotated diagram
  • For $u=\langle3,-2\rangle$ and $v=\langle-1,5\rangle$, find $u+v$ and $3u-2v$.
  • $u+v=\langle2,3\rangle$.
  • $3u-2v=\langle9,-6\rangle-\langle-2,10\rangle=\langle11,-16\rangle$.
  • The diagram separately confirms stretching and reversal under scalar multiplication.
Vectors v, two v, and negative v showing stretch and reversal
Vectors v, two v, and negative v showing stretch and reversal

Model · 3.0 minSource: Authored

You Try: perform vector operations and interpret scalar multiplication

Student practice

Let $u=\langle2,-3\rangle$ and $v=\langle-1,4\rangle$. Find $2u-3v$.

Practice · 3.0 minSource: 11

Check and Diagnose: perform vector operations and interpret scalar multiplication

Error analysis
  • Distribute the scalar to both components.
  • Check the direction of negative scalar multiples visually.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: Authored

Pacing checkpoint

Make It Stick: perform vector operations and interpret scalar multiplication

Describe how scalar sign and magnitude affect a vector.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: find vector magnitude and direction angle

Predict first

Why can a calculator’s inverse tangent answer require a quadrant adjustment?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

find vector magnitude and direction angle

Concept
  • Magnitude: $|v|=\sqrt{a^2+b^2}$.
  • Use atan2 behavior conceptually: choose the angle in the vector’s actual quadrant.

Explain · 2.5 minSource: 8, 9

Model the Skill: find vector magnitude and direction angle

Annotated diagram
  • For $v=\langle-3,4\rangle$, find magnitude and direction.
  • $|v|=\sqrt{(-3)^2+4^2}=5$.
  • The reference angle is $\tan^{-1}(4/3)\approx53.13^\circ$.
  • Quadrant II gives $\theta=180^\circ-53.13^\circ\approx126.87^\circ$.
Quadrant two vector with direction angle measured counterclockwise from the positive x-axis
Quadrant two vector with direction angle measured counterclockwise from the positive x-axis

Model · 3.0 minSource: 10

You Try: find vector magnitude and direction angle

Student practice

For $v=\langle6,3\rangle$, find magnitude and direction.

Practice · 3.0 minSource: 12

Check and Diagnose: find vector magnitude and direction angle

Error analysis
  • Check component signs before adjusting the angle.
  • Direction is measured counterclockwise from the positive x-axis.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 13

Pacing checkpoint

Make It Stick: find vector magnitude and direction angle

Write a quadrant-safe direction-angle procedure.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 14

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: Authored