Recognize and Represent Vector Quantities

Lesson 1 of 12: VM Cluster

In this lesson:

• Distinguish vectors from scalar quantities
• Draw vectors as directed line segments
• Use standard vector notation and magnitude symbols

What You Will Learn Today

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

1. Distinguish vectors from scalars
2. Represent a vector as a directed line segment
3. Identify initial point and terminal point
4. Use correct symbols for vectors and magnitudes

Which Pilot Knows Where to Go?

A flight dispatcher radios two pilots:

• Pilot A: "Fly 300 km."
• Pilot B: "Fly 300 km due east."

Which pilot knows exactly where they're going — and why?

Classify Each Quantity Before Naming

Does each need a direction to be complete?

Scalars Are Described by One Number

A scalar is fully described by its magnitude alone:

• Temperature: 72°F ✓
• Mass: 5 kg ✓
• Speed: 60 mph ✓
• Time: 3 hours ✓

No direction needed — the number is the complete description.

Vectors Require Magnitude and Direction

A vector requires both magnitude and direction:

• Wind: 25 mph due north ✓ (direction required)
• Force: 10 N inward ✓ (direction required)
• Displacement: 3 km east ✓ (direction required)

The test: speed = 60 mph (scalar, complete); velocity = 60 mph (vector, which way?)

Speed vs. Velocity: Quick Check

Speed gives you one number. Velocity gives you a number and a direction.

Drawing Vectors: From Words to Arrows

A vector carries two pieces of information — magnitude and direction. We need a geometric object that encodes both:

• Magnitude → the length of the object
• Direction → which way the object points

An arrow naturally encodes both — that's why vectors are arrows.

Anatomy of a Directed Line Segment

The arrow's length = magnitude. The arrowhead's direction = vector's direction.

Equal Vectors: Same Length, Same Direction

• Two vectors are equal if they have the same length and the same direction
• Position does not matter — the same vector can be drawn anywhere

Predict: Are These the Same Vector?

Two arrows on a coordinate grid:

• Both have length = 5 units, pointing northeast at 45°
• Arrow 1 starts at (0, 0) — Arrow 2 starts at (3, 4)

Are these the same vector, or two different vectors?

Commit to an answer before advancing.

Same Vector — Free Vectors Don't Have Positions

Same vector — identical length and identical direction.

• A vector is defined by magnitude and direction only
• Starting point is irrelevant — this is called a free vector

What makes two arrows different vectors:

• Different length → different magnitude
• Different direction → different vector

Unequal Vectors: Two Failure Cases

Change either component — magnitude or direction — and you have a different vector.

Magnitude Equals the Arrow's Length

Magnitude = the length of the directed line segment.

• Horizontal: to →
• Vertical: to →

Magnitude is always ≥ 0 — it's a length.

Example: Vector from A(1,2) to B(4,6)

Step 1: Plot A(1,2) and B(4,6); draw arrow from A to B

Step 2: Compute magnitude:

Magnitude = 5 units

Guided Practice: Vector from P(0,0) to Q(3,4)

Step 1: Draw the arrow from P to Q

Step 2: Fill in the distance formula:

This is a 3-4-5 Pythagorean triple — do you recognize the magnitude?

Free Vector: Position Does Not Matter

from P(0,0) to Q(3,4): run = 3, rise = 4, magnitude = 5.

from R(2,1) to S(5,5): run = 3, rise = 4, magnitude = 5.

Same run, same rise, same direction — same vector.

Position is not part of a vector's identity.

Three Ways to Write the Same Arrow

Three equivalent notation forms — all name the same vector:

• Bold face: — used in printed textbooks
• Arrow over letter: — used in handwriting
• Segment notation: — names the specific endpoints

All three are interchangeable.

Notation Reference: One Arrow, Three Names

One physical arrow — three ways to write it. All equivalent.

Magnitude Notation: Three Equivalent Symbols

All three are scalars — always .

Parallel: just as , magnitude measures length, never negative.

Vector or Scalar? Classify These Expressions

Vector or scalar (magnitude)?

4. (a plain number)

Classify each before advancing.

Five Notation Forms for One Arrow

Arrow from A(1,2) to B(4,6) — magnitude 5:

Identify and Correct the Notation Error

A student wrote:

"Vector points left, so ."

What's wrong? Write the correction in one sentence.

Full Procedure: From Points to Notation

Given: initial point , terminal point . No hints.

1. Draw the directed line segment; label both points
2. Write the vector in all three notation forms
3. Compute the magnitude (show all steps)
4. Is a vector or a scalar? Explain.

Key Takeaways: Vectors and Scalars

✓ Vector = magnitude + direction; scalar = magnitude only

✓ Vectors drawn as arrows: tail = initial point, tip = terminal point

✓ Equal vectors: same length AND same direction — position irrelevant (free vectors)

✓ Magnitude is a non-negative scalar (always a length)

Watch Out: Four Common Errors

Speed ≠ velocity: speed is scalar; velocity requires direction

Position doesn't change equality: same length + direction = same vector

is never negative: magnitude is a length

Vectors don't start at the origin: any starting point is valid

Next Lesson: Extracting Vector Components

Next lesson (VM.A.2): Extract a vector's direction and magnitude as two numbers — the components .

Components let you:

• Add vectors algebraically
• Find direction angles precisely
• Scale vectors by any factor