AcademyVectors In Rn
Academy
Scalar Product
Level 1 - Linear Algebra topic page in Vectors In Rn.
Principle
The scalar product, also called the dot product, takes two vectors in \(\mathbb R^n\) and returns one real number. It measures alignment: positive values mean the vectors point partly in the same direction, negative values mean they point partly in opposite directions, and zero means they are orthogonal.
Notation
\(\mathbf u\cdot\mathbf v\)
the scalar product of vectors \mathbf u and \mathbf v
\(u_i,v_i\)
matching components of \mathbf u and \mathbf v
\(|\mathbf v|\)
the length or norm of \mathbf v
\(\theta\)
the angle between two non-zero vectors
\(\mathbf u\perp\mathbf v\)
the vectors \mathbf u and \mathbf v are orthogonal
The Core Method
For vectors in \(\mathbb R^n\), multiply matching components and add the products:
Scalar product
\[\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+\cdots+u_nv_n\]
The scalar product is symmetric:
Symmetry
\[\mathbf u\cdot\mathbf v=\mathbf v\cdot\mathbf u\]
It connects components to geometry by
Geometric scalar product
\[\mathbf u\cdot\mathbf v=|\mathbf u|\,|\mathbf v|\cos\theta\]
when both vectors are non-zero. A zero scalar product is the algebraic test for orthogonality:
Orthogonality test
\[\mathbf u\perp\mathbf v\Longleftrightarrow \mathbf u\cdot\mathbf v=0\]
This test treats the zero vector carefully: \(\mathbf 0\cdot\mathbf v=0\) for every \(\mathbf v\), but the zero vector has no direction.
Worked Cases
Question
Compute \((2,-1,4)\cdot(3,5,-2)\).
Answer
Multiply matching components and add: \((2,-1,4)\cdot(3,5,-2)=2\cdot3+(-1)\cdot5+4\cdot(-2)\). Evaluate each product: \(6-5-8=-7\). Therefore the scalar product is \(-7\).
Examples
Question
What does \(\mathbf u\cdot\mathbf v<0\) tell you about two non-zero vectors?
Answer
Using \(\mathbf u\cdot\mathbf v=|\mathbf u||\mathbf v|\cos\theta\), the lengths are positive, so the sign is controlled by \(\cos\theta\). If the scalar product is negative, then \(\cos\theta<0\), so \(\theta\) is obtuse. The vectors point partly in opposite directions.