Scalar product

College/University

Definition

Also known as the dot product or inner product. It is the name given when vectors are multiplied and resulting in a scalar answer. The scalar product of two vectors at 90° (right angle) to each other will always be zero.

Worked examples

\(\vec{a} = \langle 3, 4 \rangle, \vec{b} = \langle 1, 2 \rangle \quad \Rightarrow \quad \vec{a} \cdot \vec{b} = 3(1) + 4(2) = 11\)

Multiply corresponding components and add them; the result is a scalar, not a vector.

\(\vec{u} = \langle 2, 3 \rangle, \vec{v} = \langle 6, -4 \rangle \quad \Rightarrow \quad \vec{u} \cdot \vec{v} = 2(6) + 3(-4) = 0\)

A dot product of zero tells you the vectors are perpendicular.

\(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\)

The geometric form relates the dot product to magnitudes and the angle between the vectors.

Common mistakes

\(\langle 2, 3 \rangle \cdot \langle 1, 4 \rangle = \langle 2, 12 \rangle\) → \(\langle 2, 3 \rangle \cdot \langle 1, 4 \rangle = 2(1) + 3(4) = 14\) The dot product gives a scalar, not a vector — don't multiply component-by-component and keep them as a pair.
\(\vec{a} \cdot \vec{b} = 0 \Rightarrow \vec{a} = \vec{0} \) or \( \vec{b} = \vec{0}\) → \(\vec{a} \cdot \vec{b} = 0 \Rightarrow \vec{a} \perp \vec{b} \) or one is zero\(\) A zero dot product means perpendicular vectors, not necessarily that one is the zero vector.
\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\) is false → \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\) (commutative) The dot product is commutative — order does not matter.

Where you'll use it next

You'll use the scalar product to find angles between vectors, test for perpendicularity, project one vector onto another, and compute work in physics. It appears throughout multivariable calculus and linear algebra.

Found in 1 StudyPug lesson

Mastering Dot Product in Vector Mathematics

Multivariable Calculus

Dive into the world of dot product, a fundamental concept in vector algebra. Learn its properties, applications in physics, and how it relates to vector projections and magnitudes.

UniversityUniversity