Linear Algebra Foundations
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Lesson 3 of 8

Multiply a vector by a number

Stretch, shrink or flip a vector by multiplying every component by a scalar.

The idea

An ordinary single number is called a scalar, to tell it apart from a vector. Multiplying a vector by a scalar means multiplying every component by that number.

v equals the vector with components 2, 1 3v equals the vector with components 6, 3

Each component of v was multiplied by 3: 3 × 2 = 6 and 3 × 1 = 3.

What it looks like

The name “scalar” comes from scaling. Multiplying by 3 makes the arrow three times as long, pointing the same way. Multiplying by −1 keeps the length but turns the arrow round to face the opposite way.

−6−6−5−5−4−4−3−3−2−2−1−11122334455663vv−v

Zero and negative scalars

Multiplying by 0 turns every component into 0. The result is the zero vector:

0v equals the vector with components 0, 0

A negative scalar flips the sign of every component. A negative times a positive is negative, and a negative times a negative is positive.

negative 1v equals the vector with components negative 2, negative 1

Worked example

v equals the vector with components 3, 5

Find 4v.

  1. Multiply each component

    4v equals the vector with components 4 times 3, and 4 times 5

  2. Work out each product

    4v equals the vector with components 12, 20

Common mistakes

  • Multiplying only the first component. The scalar reaches every component.
  • Adding instead of multiplying. 3v means 3 × each component, not 3 + each component.
  • Sign slips. (−2) × (−4) = 8, a positive number.

Try one

Multiply a vector by a numberPositive numbers

This first one is guided: the first hint is already open below the question.

v equals the vector with components 1, 5

Find 2v.

2v
  1. Hint 1

    The number in front, 2, is a scalar. It multiplies every component of v.

Question code vec-scale/A/21/v1.0.0