Scalar Multiplication
Scale a vector’s length.
Key idea
Scale a vector’s length.
In depth
Multiplying a vector by a scalar c scales every component, stretching by |c| and reversing direction if c is negative. It lets you build any vector along a line.
Worked example
3(1, −2) = (3, −6). −½(4, 2) = (−2, −1).
Formula
Real-world uses
- Scaling a sprite or a force
- Doubling a recipe represented as a vector of ingredients
Common mistake
Thinking scalar multiplication changes direction for any c; only a negative scalar reverses it.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Scalar Multiplication?
Scale a vector’s length. It is a topic at the intermediate level in Linear Algebra.
What is the formula for Scalar Multiplication?
The key formula is cv = (cv₁, cv₂).
Where is Scalar Multiplication used in real life?
Real-world examples: Scaling a sprite or a force; Doubling a recipe represented as a vector of ingredients.
What should I learn before Scalar Multiplication?
Make sure you are comfortable with Vector Addition, the previous topic in Linear Algebra.
What is a common mistake with Scalar Multiplication?
Thinking scalar multiplication changes direction for any c; only a negative scalar reverses it.
How can I practise Scalar Multiplication?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Scalar Multiplication?
Move on to Vector Magnitude next.
Related intermediate topics in Linear Algebra
- Scalars and Vectors – A scalar is a number; a vector has magnitude and direction.
- Vector Notation and Components – Write vectors as ordered lists of coordinates.
- Vector Addition – Add component-wise; geometrically tip-to-tail.
- Vector Magnitude – Length of a vector.
- Unit Vectors – Vectors of length 1, found by normalising.
- Matrices: Definition – A rectangular array of numbers with m rows and n columns.