Topic 4: Vectors

Introduction

Vectors can be described mathematically by using Trigonometry.

We can define a vector to be an ordered pair consisting of a magnitude and a direction. In this diagram, r is the magnitude of this vector and θ is the direction. Notice, now, that we have moved horizontally r cos(θ) and vertically r sin(θ). These are called the x-component and the y-component, respectively.

We can also write a vector conveniently in terms of the x and y component. We write ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$  for vectors. In some texts, you may see the vector written sideways, like (x, y), but when you write it will help greatly to write them downwards in columns. In print we commonly bold vectors, but since you probably don't have a pen that writes in bold print, underline your vectors, i.e. write v, or put a tilde underneath your vectors or place an arrow pointing right overtop of your vector.

Length of a vector

The distance formula can be used to find the magnitude, r, of a vector given its components with the following equation:

${\displaystyle \left\|\mathbf {a} \right\|={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}$

Where a1, a2, and a3 are the three components of the vector.

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point.

For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (displacement) vector.

Scalar multiplication

For scalar multiplication, we simply multiply each component by the scalar. We commonly use Greek letters for scalars, and Roman letters for vectors.

So for a scalar value of λ and a vector v defined by r and θ, the new vector is now λr and θ. Notice how the direction does not change.

Example

Say we have ${\displaystyle {\begin{pmatrix}2\\3\end{pmatrix}}}$  and we wish to double the magnitude. So, ${\displaystyle 2{\begin{pmatrix}2\\3\end{pmatrix}}={\begin{pmatrix}4\\6\end{pmatrix}}}$ .

Simply, to add two vectors, you must add the respective x-components together to obtain the new x-component, and likewise add the two y-components together to obtain the new y-component.

Example

Say we have ${\displaystyle \mathbf {v_{1}} ={\begin{pmatrix}2\\3\end{pmatrix}},\mathbf {v_{2}} ={\begin{pmatrix}4\\6\end{pmatrix}}}$  and we wish to add these. So, ${\displaystyle \mathbf {v_{1}} +\mathbf {v_{2}} ={\begin{pmatrix}6\\9\end{pmatrix}}}$ .

Magnitude

The magnitude of a vector is its length in R+

The dot product

The dot product of two vectors is defined as the sum of the products of the components. Symbolically we write

${\displaystyle {\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}\cdot {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}=a_{1}b_{1}+a_{2}b_{2}}$

For example,

${\displaystyle {\begin{pmatrix}3\\5\end{pmatrix}}\cdot {\begin{pmatrix}1\\-2\end{pmatrix}}=3-10=-7}$

The dot product of two vectors has an alternate form:

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

The angle θ then is important, as it shows that the dot product of two vectors is related to the angle between them. More specifically, we can calculate the dot product of two vectors - if the dot product is zero we can then say that the two vectors are perpendicular.

For example, consider simply

${\displaystyle {\begin{pmatrix}1\\1\end{pmatrix}}\cdot {\begin{pmatrix}1\\-1\end{pmatrix}}=1-1=0}$

Plot these vectors on the plane and verify for yourself that these vectors are perpendicular.

Vectors of 3D-lines

Cartesian equation: ${\displaystyle {\tfrac {x-a}{l}}={\tfrac {y-b}{m}}={\tfrac {z-c}{n}}}$ . where a, b, and c are the coordinates on the vector line.

Vector equation: r = ${\displaystyle {\begin{pmatrix}a\\b\\c\end{pmatrix}}}$