Vectors/Vector Algebra

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are vectors, then the sum ${\displaystyle \mathbf {c} =\mathbf {a} +\mathbf {b} \,}$ is also a vector.

The two vectors can also be subtracted from one another to give another vector ${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {b} \,}$.

Multiplication of a vector ${\displaystyle \mathbf {b} \,}$ by a scalar ${\displaystyle \lambda \,}$ has the effect of stretching or shrinking the vector.

You can form a unit vector ${\displaystyle {\hat {\mathbf {b} }}\,}$ that is parallel to ${\displaystyle \mathbf {b} \,}$ by dividing by the length of the vector ${\displaystyle |\mathbf {b} |\,}$. Thus,

${\displaystyle {\hat {\mathbf {b} }}={\frac {\mathbf {b} }{|\mathbf {b} |}}~.}$

The scalar product or inner product or dot product of two vectors is defined as

${\displaystyle \mathbf {a} \cdot \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\cos(\theta )}$

where ${\displaystyle \theta \,}$ is the angle between the two vectors (see Figure 2(b)).

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are perpendicular to each other, ${\displaystyle \theta =\pi /2\,}$ and ${\displaystyle \cos(\theta )=0\,}$. Therefore, ${\displaystyle {\mathbf {a} }\cdot {\mathbf {b} }=0}$.

The dot product therefore has the geometric interpretation as the length of the projection of ${\displaystyle \mathbf {a} \,}$ onto the unit vector ${\displaystyle {\hat {\mathbf {b} }}\,}$ when the two vectors are placed so that they start from the same point (tail-to-tail).

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as

${\displaystyle {\mathbf {a} }\cdot {\mathbf {b} }=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=\sum _{i=1..3}a_{i}b_{i}~.}$

If the vector is ${\displaystyle n}$ dimensional, the dot product is written as

${\displaystyle {\mathbf {a} }\cdot {\mathbf {b} }=\sum _{i=1..n}a_{i}b_{i}~.}$

Using the Einstein summation convention, we can also write the scalar product as

${\displaystyle {\mathbf {a} }\cdot {\mathbf {b} }=a_{i}b_{i}~.}$

Also notice that the following also hold for the scalar product

1. ${\displaystyle {\mathbf {a} }\cdot {\mathbf {b} }={\mathbf {b} }\cdot {\mathbf {a} }}$ (commutative law).
2. ${\displaystyle {\mathbf {a} }\cdot {(\mathbf {b} +\mathbf {c} )}={\mathbf {a} }\cdot {\mathbf {b} }+{\mathbf {a} }\cdot {\mathbf {c} }}$ (distributive law).

The vector product (or cross product) of two vectors ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ is another vector ${\displaystyle \mathbf {c} \,}$ defined as

${\displaystyle \mathbf {c} ={\mathbf {a} }\times {\mathbf {b} }=|\mathbf {a} ||\mathbf {b} |\sin(\theta ){\hat {\mathbf {c} }}}$

where ${\displaystyle \theta \,}$ is the angle between ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$, and ${\displaystyle {\hat {\mathbf {c} }}\,}$ is a unit vector perpendicular to the plane containing ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ in the right-handed sense.

In terms of the orthonormal basis ${\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,}$, the cross product can be written in the form of a determinant

${\displaystyle {\mathbf {a} }\times {\mathbf {b} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}.}$

In index notation, the cross product can be written as

${\displaystyle {\mathbf {a} }\times {\mathbf {b} }\equiv e_{ijk}a_{j}b_{k}.}$

where ${\displaystyle e_{ijk}}$ is the Levi-Civita symbol (also called the permutation symbol, alternating tensor).

Some useful vector identities are given below.

1. ${\displaystyle {\mathbf {a} }\times {\mathbf {b} }=-{\mathbf {b} }\times {\mathbf {a} }}$.
2. ${\displaystyle {\mathbf {a} }\times {\mathbf {b} +\mathbf {c} }={\mathbf {a} }\times {\mathbf {b} }+{\mathbf {a} }\times {\mathbf {c} }}$.
3. ${\displaystyle {\mathbf {a} }\times {({\mathbf {b} }\times {\mathbf {c} })}=\mathbf {b} ({\mathbf {a} }\cdot {\mathbf {c} })-\mathbf {c} ({\mathbf {a} }\cdot {\mathbf {b} })}$
4. ${\displaystyle {({\mathbf {a} }\times {\mathbf {b} })}\times {\mathbf {c} }=\mathbf {b} ({\mathbf {a} }\cdot {\mathbf {c} })-\mathbf {a} ({\mathbf {b} }\cdot {\mathbf {c} })}$
5. ${\displaystyle {\mathbf {a} }\times {\mathbf {a} }=\mathbf {0} }$
6. ${\displaystyle {\mathbf {a} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}={\mathbf {b} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}=\mathbf {0} }$
7. ${\displaystyle {({\mathbf {a} }\times {\mathbf {b} })}\cdot {\mathbf {c} }={\mathbf {a} }\cdot {({\mathbf {b} }\times {\mathbf {c} })}}$

For more details on the topics of this chapter, see Vectors in the wikibook on Calculus.