Find Components of a Vector

Lesson 2 of 12: VM Cluster

In this lesson:

• Compute components by subtracting initial from terminal coordinates
• Express vectors in angle-bracket notation
• Compute magnitude from components

What You Will Learn Today

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

1. Find components by subtracting initial-point coordinates from terminal-point coordinates
2. Express a vector in angle-bracket notation
3. Interpret components as signed horizontal and vertical displacements
4. Compute magnitude using

Recall: Directed Line Segments from Lesson 1

• Initial point: the tail of the arrow — where the vector starts
• Terminal point: the arrowhead — where the vector ends
• Direction: the arrow points from initial → terminal
• Equal vectors: same length and direction, regardless of position

Can you locate the initial point, terminal point, and direction of a drawn arrow?

What Two Numbers Describe This Move?

A warehouse robot receives a movement command:

"Move from shelf A(1, 3) to shelf B(4, 7)."

To program this move, the computer needs numbers — not a picture.

What two numbers describe this movement completely?

Count Grid Squares, Then Use Subtraction

Component Formula: Terminal Minus Initial

• — terminal minus initial
• — terminal minus initial

Memory cue: terminal minus initial — the vector points toward terminal.

Components Don't Depend on Starting Position

• From to :
• From to :

Worked Examples: Two Component Computations

Example 1: From to

Example 2: From to

Check: example 2 moved right (+) and down (−) — signs match.

Check-In: Apply the Formula Yourself

Vector from to :

Apply the formula:

Check the sign of each component against the direction of travel.

From Number Pair to Angle-Bracket Notation

• Angle brackets signal: this is a displacement (vector)
• Parentheses signal: this is a position (point)

— similar notation, completely different meanings.

Reading Direction from Component Signs

• Positive : moves right — Negative : moves left
• Positive : moves up — Negative : moves down
• Zero component: no movement along that axis

→ right+up; → left+down.

Four Displacement Directions from Sign Patterns

Every component pair maps to one of four displacement quadrants.

When One Component Is Zero

• : no horizontal movement — straight up, magnitude 5
• : no vertical movement — straight left, magnitude 4
• : no movement at all — the zero vector

When one component is zero, the vector lies along an axis.

Check-In: Sketch a Component Vector

Sketch on a coordinate grid.

Describe its direction in plain words.

What quadrant of displacement does it fall in?

Predict: What Happens When You Reverse Direction?

Consider the vector .

Now reverse the direction — the arrow points the opposite way.

Predict: What are the components of the reversed vector?

Commit to an answer before the next slide.

Components Form a Right Triangle for Magnitude

• Move 3 right (horizontal leg)
• Move 4 up (vertical leg)
• The arrow is the hypotenuse

Magnitude Formula: Pythagorean Theorem Applied

By the Pythagorean theorem on the component right triangle:

Key fact: Squaring removes the sign — .

Magnitude is always — it's a length, not a displacement.

Computing Magnitude: Three Worked Examples

1: (3-4-5 triple)

2: (5-12-13 triple)

3: (irrational — leave in radical form)

Guided Practice: Magnitude of

Fill in the magnitude formula:

Does the negative component affect the final magnitude?

Find the Displacement vs. Position Error

A student computed the following:

"Vector from to : the component form is ."

What did this student do wrong?

Write the correct component form and explain the difference.

End-to-End Practice: No Scaffold Given

From to . Complete all three tasks:

1. Write the component form
2. State which quadrant of displacement this falls in
3. Compute the magnitude — leave in simplified form if needed

Key Takeaways: Components and Magnitude

✓ Component formula: — terminal minus initial

✓ Signs encode direction: positive = right/up; negative = left/down

✓ Magnitude formula: — Pythagorean theorem on components

✓ Position-independence: same displacement = same vector, any starting point

Watch Out: Four Errors to Avoid

Subtract terminal minus initial — reversed order flips the direction

— angle brackets signal displacement, not position

Hypotenuse, not sum: , not

Squaring removes sign: has the same magnitude as

Next Lesson: Adding Vectors by Components

You can now express any vector as and find its magnitude.

Next (VM.B.4): Add two vectors by adding their components:

The component form you built today is the engine for all vector arithmetic.