Addition and Subtraction

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are vectors, then the sum ${\displaystyle \mathbf {c} =\mathbf {a} +\mathbf {b} \,}$ is also a vector (see Figure 2(a)).

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

Multiplication by a scalar

Multiplication of a vector ${\displaystyle \mathbf {b} \,}$ by a scalar ${\displaystyle \lambda \,}$ has the effect of stretching or shrinking the vector (see Figure 2(b)).

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

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

Scalar product of two vectors

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.

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).

Vector product of two vectors

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 (see Figure 3 for a geometric interpretation)

Figure 3: Vector product of two vectors.

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). This latter expression is easy to remember if you recognize that xyz, yzx, and zxy are "positive" and the others are negative: xzy, yxz, zyx.

If ${\displaystyle \mathbf {c} ={\mathbf {a} }\times {\mathbf {b} }}$, then

${\displaystyle c_{x}=a_{y}b_{z}-a_{z}b_{y}}$

${\displaystyle c_{y}=a_{z}b_{x}-a_{x}b_{z}}$

${\displaystyle c_{z}=a_{x}b_{y}-a_{y}b_{z}}$

Identities from Vector Algebra

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. ${\displaystyle {({\mathbf {a} }\times {\mathbf {b} })}\cdot {\mathbf {c} }={\mathbf {a} }\cdot {({\mathbf {b} }\times {\mathbf {c} })}~}$

The rest of this resource has been moved to Vector calculus.

Real Number Coordination

Scalar Value

${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle \theta ={\frac {y}{x}}}$

${\displaystyle x(\theta )=rCos\theta }$

${\displaystyle y(\theta )=rSin\theta }$

Vector

${\displaystyle r(\theta )=x(\theta )+y(\theta )=r(Cos\theta +Sin\theta )}$

Vector composites

${\displaystyle \nabla \cdot r(\theta )=x(\theta )=rCos\theta }$

${\displaystyle \nabla \times r(\theta )=y(\theta )=rSin\theta }$

Complex Number Coordination

Complex Number

Scalar Value

${\displaystyle z={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle \theta ={\frac {y}{x}}}$

${\displaystyle x(\theta )=zCos\theta }$

${\displaystyle jy(\theta )=jzSin\theta }$

Vector

${\displaystyle r(\theta )=x(\theta )+jy(\theta )=r(Cos\theta +jSin\theta )}$

Vector composites

${\displaystyle \nabla \cdot z(\theta )=x(\theta )=zCos\theta ={\frac {z+z^{*}}{2}}}$

${\displaystyle \nabla \times z(\theta )=jy(\theta )=jzSin\theta ={\frac {z-z^{*}}{2j}}}$

Complex Conjugate Number

Scalar Value

${\displaystyle z={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle \theta =-{\frac {y}{x}}}$

${\displaystyle x(\theta )=zCos\theta }$

${\displaystyle -jy(\theta )=-jzSin\theta }$

Vector

${\displaystyle r(\theta )=x(\theta )-jy(\theta )=r(Cos\theta -jSin\theta )}$

Vector composites

${\displaystyle \nabla \cdot z(\theta )=x(\theta )=zCos\theta ={\frac {z+z^{*}}{2}}}$

${\displaystyle \nabla \times z(\theta )=-jy(\theta )=-jzSin\theta ={\frac {z^{*}-z}{2j}}}$

x,y,z an r vectors

Commonly in physics, we use position vectors to describe where something is in the space we are considering, or how its position is changing at that moment in time. Position vectors are written as summations of scalars multiplied by unit vectors. For example:

${\displaystyle {\vec {r}}=x{\hat {i}}+y{\hat {j}}+z{\hat {k}}}$

where x, y and z are scalars and ${\displaystyle {\hat {i}},{\hat {j}}}$ and ${\displaystyle {\hat {k}}}$ are unit vectors of the Cartesian (René Descartes) coordinate system. A unit vector is a special vector which has magnitude 1 and points along one of the coordinate frame's axes. Unit vectors for each direction can be written as either ${\displaystyle ({\hat {i}},{\hat {j}},{\hat {k}})}$ or ${\displaystyle ({\hat {x}},{\hat {y}},{\hat {z}})}$. The figure to the left illustrates this with ${\displaystyle 3{\hat {x}}+2{\hat {y}}}$. A vector itself is typically indicated by either an arrow: ${\displaystyle {\vec {v}}}$, or just by boldface type: v, so the vector above as a complete equation would be denoted as:

${\displaystyle {\vec {v}}=v_{x}{\hat {i}}+v_{y}{\hat {j}}+v_{z}{\hat {k}}}$

This velocity vector follows the convention that subscripts denote the components ${\displaystyle (v_{x},v_{y},v_{z})}$ of the velocity vector. Writing the components of ${\displaystyle {\vec {r}}}$ as ${\displaystyle (r_{x},r_{y},r_{z})}$ would be more consistent but is almost never done.

You can find the magnitude of a vector with this formula ${\displaystyle |{\vec {v}}|={\sqrt {\sum _{i}(x_{i}^{2})}}}$. For example, in two-dimensional space, this equation reduces to:

${\displaystyle |{\vec {v}}|={\sqrt {x^{2}+y^{2}}}}$.

For three-dimensional space, this equation becomes:

${\displaystyle |{\vec {v}}|={\sqrt {x^{2}+y^{2}+z^{2}}}}$.