Multiply a Vector by a Scalar

Lesson 5 of 12: VM Cluster

In this lesson:

• Scale a vector graphically — stretch, shrink, or reverse
• Compute scalar multiplication component-wise
• Find the magnitude and direction of a scalar multiple; construct unit vectors

What You Will Learn Today

1. Draw : scale length by , flip when
2. Compute
3. Find
4. State direction: same as v when ; opposite when
5. Build

Scaling a Force: What Changes?

A force pushes a box with 10 N in a fixed direction.

• What does 2 × that force look like?
• What does 0.5 × that force look like?
• What does −1 × that force look like?

The direction of a force and its magnitude are separate properties.

Seeing Scalar Multiples on a Grid

How Scalar Multiplication Transforms a Vector

For vector v and real number :

• = new magnitude
• Same direction when ; reversed when
• All scalar multiples lie along the same line as v

Scaling stretches or shrinks, and may flip. It never rotates.

Special Values of the Scalar

• c = 0: — the zero vector; no direction defined
• c = 1: — identity; unchanged
• c = −1: — additive inverse; same length, reversed

These are the three boundary cases: collapse, preserve, and reverse.

Check-In: Draw Four Scalar Multiples Now

Vector .

On a coordinate grid:

1. Draw v
2. Draw — predict the endpoint before drawing
3. Draw
4. Draw

Predict the Component Form First

v =

Before the rule: predict in component form.

What happens to each component when the whole vector doubles?

Component Rule: Multiply Each Entry

Each component multiplies by the scalar independently.

Worked Examples: Positive and Negative Scalars

• — triple length, same direction
• — double length, reversed

Why the Component Rule Works

The component rule scales both projections equally:

• Horizontal projection:
• Vertical projection:

The ratio is unchanged → direction preserved for c > 0

Scaling uniformly in all directions is the algebraic definition of "not rotating."

Predict the Magnitude of 3v

v = , so .

Predict: What is ?

If the arrow is three times as long, the magnitude should be...

Magnitude Formula: Derivation from Components

Key step: , not — magnitude is always .

Direction Rule for Scalar Multiples

When (i.e., neither nor ):

• : points along v (same direction)
• : points against v (opposite direction)

The sign of c controls direction; controls magnitude.

Unit Vectors Isolate Direction from Magnitude

Worked Example: Finding a Unit Vector

v = ,

Verify: ✓

Worked Example: Negative Scalar with Magnitude

v = , compute and its magnitude.

Direction: opposite to v — confirmed by sign-flipped components.

Find and Fix the Magnitude Error

A student wrote:

", scalar : magnitude of = ."

What is wrong? Write the correct answer and explain.

Full Practice: All Four Skills Together

Given v = . No scaffold.

1. Find
2. Find
3. State the direction of relative to v
4. Find the unit vector

Key Takeaways from Scalar Multiplication

✓ Scale: times the length; direction same when , reversed when

✓ Component rule:

✓ Magnitude: — use , not

✓ Unit vector: — magnitude 1, same direction

Next Lesson: Scaling Entire Matrices

You can now multiply any vector by any scalar.

Next (VM.C.7): Multiply a matrix by a scalar:

The same component-wise rule — applied to every entry of the matrix.