Scalar

High School

Definition

Any real number used in linear algebra. A quantity with a magnitude but no direction. Mass, height, temperature, distance are all examples of scalar quantities.

Worked examples

\(3\vec{v}\)

The scalar \(3\) scales the vector \(\vec{v}\) by stretching it three times longer in the same direction.

\(k = -2.5\)

A scalar like \(-2.5\) is just a real number; it has size but no direction, unlike a vector.

Common mistakes

\(5\) is a vector because it can be written as \(\langle 5 \rangle\) → \(5\) is a scalar; vectors need direction, e.g. \(\langle 5, 0 \rangle\) in 2D A single number alone has no direction component and is always a scalar.
Temperature \(20^\circ\)C\(\) is a vector because it points up on a thermometer → Temperature is a scalar; it has magnitude only, no spatial direction Physical direction (up/down/left/right in space) is needed for a vector; scale readings are scalar.

Where you'll use it next

Scalars are the building blocks for vector operations and matrix equations in linear algebra, and they reappear in calculus for scaling functions, in physics for mass and energy, and throughout engineering and data science.

Found in 1 StudyPug lesson

Scalar Multiplication of Vectors

Grade 12 Math

Discover how multiplying a vector by a scalar stretches, shrinks, or flips it, and practice the component formula with step-by-step examples.

12th Grade12th