Vectors Geometry Topic

Vectors

This picture is an example of position vector (it starts at the origin)

Vector is a quantity which has a magnitude and a direction.
Magnitude of a 3D position vector can be calculated as follows \(a=\sqrt{x^2 + y^2 + z^2}\)
This is its notation: \(\vec{a}\)
Column representation:
\(\vec{a}=\begin{pmatrix} 1 \\ 3 \end{pmatrix}\)
The components of a 2D vector can be expressed like this:
\(x = cos\theta\)
\(y = sin\theta\)
A vector does not have a to have a position
Direction vectors do not start at the origin and connect two points (first point or a position vector is their starting position)

Vector addition is commutative, substraction is not
\(\vec{a}+\vec{b}=\begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix}\)

Scalar Product

It is also called the dot product
The results describes how well two vectors travel together or how parallel they are
It is commutative

\({\vec{a}}\cdot{\vec{b}}=ab\cos\theta\) (\(a\) and \(b\) are just magnitudes)
\(\mathbb{R_2}\)
\(\vec{a}\cdot{\vec{b}}=a_1 a_2 + b_1 b_2\)
The result is a scalar as the names implies

Scalar value:
- is the largest when the vectors are parallel, \(cos(0,360)=1\)
- is the smallest when they are anti-parallel, \(cos(180)=-1\)
- is 0 when they are perpendicular, \(cos(90,270)=0\)

The dot product can tell us the angle between two vectors

\(\theta=arcos(\frac{{\vec{a}\cdot{\vec{b}}}}{ab})\)

Vector Product

The result is a vector, so it includes a direction, as the name implies
It is also called the cross product because of the operator
It is non-commutative
It equals to \(\vec{0}\) if the vectors are parallel
\({\vec{a}}\times{\vec{b}}=-({\vec{b}}\times{\vec{a}})\)

It can be written in the matrix determinant form
The top row are unit vectors (vectors of magnitude 1)
\({\vec{A}}\times{\vec{B}}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}\)

Converting the determinant form into the standard form

\(\left(\begin{array}{cccc}\color{Green}{+}&\color{blue}{-}&\color{Green}{+}&\color{blue}{-} \\ \color{blue}{-}&\color{Green}{+}&\color{blue}{-}&\color{Green}{-} \\ \color{Green}{+}&\color{blue}{-}&\color{Green}{+}&\color{blue}{-} \\ \color{blue}{-}&\color{Green}{+}&\color{blue}{-}&\color{Green}{+}\end{array}\right)\) This pattern indicates the polarity of the unit vector (if we multiply it by 1 or -1)

We multiply the unit vector by the difference of the product of the left-to-right diagonal and the product of the right-to-left diagonal (Refer to the picture on how your form the matrix where we get these diagonals from)
\({\vec{A}}\times{\vec{B}}=\vec{i}(A_{y}B_{z}-A_{z}B_{y})-\vec{j}(A_{x}B_{z}-A_{z}B_{x})+\vec{k}(A_{x}B_{y}-A_{y}B_{x})\)

The resulting vector is perpendicular to the parallelogram formed by the two vectors
The right hand rule determines whether it will point upwards or downwards from the parallelogram
A vector perpendicular to a plane or another vector is the normal vector
\(|{\vec{a}}\times{\vec{b}}|=ab\sin\theta\)
The scalar of the vector product is the area of the parallelogram formed by the two vectors

Using the cross product to the get the minimum distance from a 'base' vector to a certain point

\(|\vec{a}|\sin\theta=h_b\)
\(|{\vec{a}}\times{\vec{b}}|=(Length of Base)(Vertical Height)=|\vec{b}|h_b\)
\(\frac{|{\vec{a}}\times{\vec{b}}|}{|\vec{b}|}=h_b\)

Topic 19: Vectors

Vectors

Scalar Product

Vector Product