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

Grade 12 Math

We have learnt that for a vector arrow, the greater the length, the greater the magnitude. Now what if we somehow want to increase or decrease the magnitude of an existing vector? In this section, we will introduce scalar multiplication – a tool that allows us to lengthen or shorten a vector arrow, in other words, a technique that alters the magnitude of a vector.

12th Grade12th