Vectors/Vector Algebra

Two vectors are added head to tail.

Vector Algebra Operations

Addition and Subtraction

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

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

Multiplication by a scalar

Multiplicaton by 2 doubles the length of a vector

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

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

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

Scalar product of two vectors

The scalar product depends on the cosine of the angle between two vectors.

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

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

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

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

The dot product therefore has the geometric interpretation as the length of the projection of $\mathbf {a} \,$ onto the unit vector ${\hat {\mathbf {b} }}\,$ 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

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

If the vector is $n$ dimensional, the dot product is written as

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

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

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

Also notice that the following also hold for the scalar product

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

Vector product of two vectors

The area of the parallelogram generated by two vectors is the length of their cross product

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

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

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

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

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

In index notation, the cross product can be written as

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

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

Identities from Vector Algebra

Some useful vector identities are given below.

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

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