AcademyMeasurement and Vectors
Academy
Dot and Cross Products
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Dot and cross products measure parallel alignment and perpendicular oriented area.
Notation
\(\vec{a}\cdot\vec{b}\)
scalar product
a unit times b unit
\(\vec{a}\times\vec{b}\)
vector product
a unit times b unit
\(\theta\)
angle from a to b
rad or deg
\(\hat{n}\)
right-hand-rule direction
none
\(a_{\parallel}\)
component of a parallel to b
same as a
Method
The angle between the vectors controls both products: cosine keeps the parallel part, sine keeps the perpendicular spread.
Parallel part
\[a_{\parallel}=|\vec{a}|\cos\theta\]
Dot product
\[\vec{a}\cdot\vec{b}=a_{\parallel}|\vec{b}|\]
Perpendicular part
\[|\vec{a}_{\perp}|=|\vec{a}|\sin\theta\]
Cross magnitude
\[|\vec{a}\times\vec{b}|=|\vec{a}|\,|\vec{b}|\sin\theta\]
Orientation
\[\vec{a}\times\vec{b}=|\vec{a}|\,|\vec{b}|\sin\theta\,\hat{n}\]
Rules
Dot geometry
\[\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\]
Dot components
\[\vec{a}\cdot\vec{b}=a_xb_x+a_yb_y+a_zb_z\]
Cross geometry
\[\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\]
Cross magnitude
\[|\vec{a}\times\vec{b}|=\text{area of parallelogram}\]
Checks
- Dot product is scalar.
- Cross product is vector.
- Parallel vectors have zero cross product.
- Perpendicular vectors have zero dot product.