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Products of Vectors

The Dot Product (Scalar Product)

Dot Product

Let $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ . The dot product is

\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3

The result is a scalar, hence the name scalar product.

Geometric Meaning

Geometric meaning of the dot product

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

where $\theta$ is the angle between the two vectors.

When $\theta = 0^\circ$ : $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|$
When $\theta = 90^\circ$ : $\vec{a} \cdot \vec{b} = 0$
When $\theta = 180^\circ$ : $\vec{a} \cdot \vec{b} = -|\vec{a}| |\vec{b}|$

Properties

Properties of the dot product

Commutativity: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

Distributivity: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Scalar association: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$

Self inner product: $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Applications

Perpendicularity test:

\vec{a} \perp \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b} = 0

Angle formula:

\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}

The Cross Product (Vector Product)

Cross Product

Let $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ . The cross product is

\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)

The result is a vector, hence the name vector product.

Determinant Formula

Determinant representation of the cross product

\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Geometric Meaning

Geometric meaning of the cross product

The result vector is perpendicular to the plane of $\vec{a}$ and $\vec{b}$ , with direction given by the right-hand rule:

|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta

Right-hand rule: curl the fingers of your right hand from $\vec{a}$ toward $\vec{b}$ ; your thumb points in the direction of $\vec{a} \times \vec{b}$ .

Properties

Properties of the cross product

Anti-commutativity: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$

Distributivity: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

Scalar association: $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$

Self cross product: $\vec{a} \times \vec{a} = \vec{0}$

Applications

Parallelism test:

\vec{a} \parallel \vec{b} \Leftrightarrow \vec{a} \times \vec{b} = \vec{0}

Area: the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ is $|\vec{a} \times \vec{b}|$ .

The Scalar Triple Product

Scalar Triple Product

For vectors $\vec{a}, \vec{b}, \vec{c}$ in 3-space, the scalar triple product is

[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Its absolute value equals the volume of the parallelepiped spanned by the three vectors.

Coplanarity test: three vectors are coplanar if and only if their scalar triple product is zero:

[\vec{a}, \vec{b}, \vec{c}] = 0

Worked Example: Mixed Product

Find the scalar triple product of $\vec{a}=(1,0,0)$ , $\vec{b}=(0,1,0)$ , $\vec{c}=(0,0,1)$ .

Solution: $\vec{b}\times\vec{c} = (1,0,0)$ , so $[\vec{a},\vec{b},\vec{c}] = \vec{a}\cdot(1,0,0) = 1$ . The three unit vectors span a unit cube of volume 1.

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$\vec{a} \cdot \vec{b}$	scalar	a dot b	The dot product of $\vec{a}$ and $\vec{b}$
$\vec{a} \times \vec{b}$	vector	a cross b	The cross product of $\vec{a}$ and $\vec{b}$
$[\vec{a}, \vec{b}, \vec{c}]$	scalar	triple product	The scalar triple product
$\theta$	Greek letter	theta	The angle between two vectors
$\vec{i}, \vec{j}, \vec{k}$	vectors	i, j, k	The standard unit vectors along the axes

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
dot product	数量积	/dɒt ˈprɒdʌkt/	The scalar product $\vec{a}\cdot\vec{b}$
cross product	向量积	/krɒs ˈprɒdʌkt/	The vector product $\vec{a}\times\vec{b}$
scalar triple product	混合积	/ˈskeɪlə ˈtrɪpəl ˈprɒdʌkt/	$\vec{a}\cdot(\vec{b}\times\vec{c})$
perpendicular	垂直	/ˌpɜːpənˈdɪkjʊlə/	At right angles
parallel	平行	/ˈpærəlel/	Having the same direction
right-hand rule	右手定则	/raɪt hænd ruːl/	Rule for the direction of a cross product
parallelogram	平行四边形	/ˌpærəˈleləɡræm/	A four-sided figure with opposite sides parallel
parallelepiped	平行六面体	/ˌpærəlelˈepɪped/	A 3-D figure of three parallelograms

Basic Concepts and Operations of Vectors

Learn the definition, linear operations, coordinate representation, and linear combinations of vectors.

Analytic Geometry of Space

Learn the equations of points, lines, and planes in space, their positional relationships, and distance and angle formulas.