Vector Dot Product and Cross Product Formulae

This lesson is a reference of the main formulae for the scalar (dot) product and the vector (cross) product of two vectors, with the results for angles, projections, and areas.

Scalar (dot) product

For two vectors $\vec a$ and $\vec b$ with angle $\theta$ between them, the dot product is

$\vec a \cdot \vec b = |\vec a|\,|\vec b|\cos\theta.$

The result is a real number (a scalar). Two non-zero vectors are perpendicular exactly when their dot product is zero:

$\vec a \cdot \vec b = 0 \iff \vec a \perp \vec b.$

Special angles. If $\theta = 0$ then $\vec a \cdot \vec b = |\vec a|\,|\vec b|$. If $\theta = \pi$ then $\vec a \cdot \vec b = -|\vec a|\,|\vec b|$. A vector dotted with itself gives its squared length:

$\vec a \cdot \vec a = |\vec a|^2.$

Dot products of the unit vectors

$\hat\imath \cdot \hat\imath = \hat\jmath \cdot \hat\jmath = \hat k \cdot \hat k = 1,$

$\hat\imath \cdot \hat\jmath = \hat\jmath \cdot \hat k = \hat k \cdot \hat\imath = 0.$

Angle between two vectors

Rearranging the definition gives the cosine of the angle:

$\cos\theta = \frac{\vec a \cdot \vec b}{|\vec a|\,|\vec b|}, \qquad \theta = \cos^{-1}\!\left(\frac{\vec a \cdot \vec b}{|\vec a|\,|\vec b|}\right).$

Properties of the dot product

• Commutative: $\vec a \cdot \vec b = \vec b \cdot \vec a$.
• Distributive: $\vec a \cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c$.
• Scalar multiple: $\lambda(\vec a \cdot \vec b) = (\lambda\vec a)\cdot \vec b = \vec a \cdot (\lambda\vec b)$.

Dot product from components

If $\vec a = a_1\hat\imath + a_2\hat\jmath + a_3\hat k$ and $\vec b = b_1\hat\imath + b_2\hat\jmath + b_3\hat k$, then

$\vec a \cdot \vec b = a_1 b_1 + a_2 b_2 + a_3 b_3.$

Projection

The projection of $\vec a$ on $\vec b$, and of $\vec b$ on $\vec a$, are

$\text{proj of }\vec a\text{ on }\vec b = \frac{\vec a \cdot \vec b}{|\vec b|}, \qquad \text{proj of }\vec b\text{ on }\vec a = \frac{\vec a \cdot \vec b}{|\vec a|}.$

When $\theta = \pi/2$ (or $3\pi/2$) the projection is $0$.

Direction cosines and unit vector

For $\vec a = a_1\hat\imath + a_2\hat\jmath + a_3\hat k$, the direction cosines are

$\cos\alpha = \frac{a_1}{|\vec a|}, \quad \cos\beta = \frac{a_2}{|\vec a|}, \quad \cos\gamma = \frac{a_3}{|\vec a|},$

so that

$\hat a = \cos\alpha\,\hat\imath + \cos\beta\,\hat\jmath + \cos\gamma\,\hat k.$

The magnitude is the square root of the sum of the squares of the components,

$|\vec a| = \sqrt{a_1^2 + a_2^2 + a_3^2},$

and the unit vector in the direction of $\vec a$ is

$\hat a = \frac{\vec a}{|\vec a|}.$

Vector (cross) product

The cross product of $\vec a$ and $\vec b$ is a vector,

$\vec a \times \vec b = |\vec a|\,|\vec b|\sin\theta\;\hat n,$

where $\hat n$ is the unit vector perpendicular to both $\vec a$ and $\vec b$. The vectors are parallel exactly when the cross product is the zero vector:

$\vec a \times \vec b = \vec 0 \iff \vec a \parallel \vec b.$

In particular $\vec a \times \vec a = \vec 0$.

Cross products of the unit vectors

$\hat\imath \times \hat\imath = \hat\jmath \times \hat\jmath = \hat k \times \hat k = \vec 0,$

$\hat\imath \times \hat\jmath = \hat k, \quad \hat\jmath \times \hat k = \hat\imath, \quad \hat k \times \hat\imath = \hat\jmath,$

$\hat\jmath \times \hat\imath = -\hat k, \quad \hat k \times \hat\jmath = -\hat\imath, \quad \hat\imath \times \hat k = -\hat\jmath.$

Angle from the cross product

$\sin\theta = \frac{|\vec a \times \vec b|}{|\vec a|\,|\vec b|}.$

Areas

For a triangle with adjacent sides $\vec a$ and $\vec b$,

$\text{Area} = \tfrac{1}{2}\,|\vec a \times \vec b|.$

For a parallelogram with adjacent sides $\vec a$ and $\vec b$,

$\text{Area} = |\vec a \times \vec b|.$

Properties of the cross product

• Distributive: $\vec a \times (\vec b + \vec c) = \vec a \times \vec b + \vec a \times \vec c$.
• Scalar multiple: $\lambda(\vec a \times \vec b) = (\lambda\vec a)\times\vec b = \vec a \times (\lambda\vec b)$.

Cross product from components

If $\vec a = a_1\hat\imath + a_2\hat\jmath + a_3\hat k$ and $\vec b = b_1\hat\imath + b_2\hat\jmath + b_3\hat k$, then

$\vec a \times \vec b = \begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}.$

Key takeaways

• The dot product gives a scalar and is zero exactly when the vectors are perpendicular; the cross product gives a vector and is zero exactly when they are parallel.
• Angles come from $\cos\theta = \dfrac{\vec a \cdot \vec b}{|\vec a|\,|\vec b|}$ and $\sin\theta = \dfrac{|\vec a \times \vec b|}{|\vec a|\,|\vec b|}$.
• The cross product gives areas: half its magnitude for a triangle, its full magnitude for a parallelogram.