11 Vectors

A scalar is a quantity that has magnitude. Scalars include quantities like length, mass, time and speed.

A vector is a quantity that has magnitude and direction. Vectors include quantities like displacement, weight, force and velocity. Vectors may be represented as $\vec{a}$, $\vec{b}$ and $\vec{c}$ or more simply as $\textbf{a, b }$ and $ \textbf{c} $.

We will start by looking at vectors in two dimensions.

11.1 Unit Vectors

Vector $\textbf{a}$ has scalar components $a_x$ in the $x$ direction and $a_y$ in the $y$ direction.

Along the $x$-axis there is a vector which is one unit long called $i$ and along the $y$-axis there is a vector which is one unit long called $j$ so we can write $\textbf{a} = a_xi + a_yj$

The magnitude of $\textbf{a}$ can be found by $|a| = \sqrt{a_x^2+a_y^2} $.

Vector with unit vectors
Fig 11.1 - Vector with unit vectors

11.2 2D Direction Cosines

The angle between the vector $\textbf{a}$ and the $x$-axis is given by $\alpha=cos^{-1}(a_x/|a|)$. The angle between the vector $\textbf{a}$ and the $y$-axis is given by $\beta=cos^{-1}(a_y/|a|)$. These angles are called direction cosines. Obviously $\alpha+\beta=\pi/2$. These angles become more useful when we work with 3D vectors.

The direction cosines are often refered to as $l$ and $m$ where $l=cos(\alpha)$ and $m=cos(\beta)$.

$l=cos(\alpha)=a_x/|a|$
$m=cos(\beta)=a_y/|a|$

11.3 Vector Addition

Vector Addition
Fig 11.3 - Vector Addition

We can add two vectors, $\textbf{a}$ and $\textbf{b}$ by adding their respective components.

$\textbf{a} = a_xi + a_yj$ and $\textbf{b} = b_xi + b_yj$

so $\textbf{a} + \textbf{b} = (a_x + b_x)i + (a_y + b_y)j$

Example 11.3: Given $\textbf{a}=3i+4j$ and $\textbf{b}=2i+6j$ find $\textbf{a}+\textbf{b}$.

$\textbf{a}$=$3i+4j$
$\textbf{b}$=$2i+6j$
$\textbf{a}+\textbf{b}$=$(3+2)i+(4+6)j$
 =$5i+10j$

11.4 Vector Subtraction

We can subtract vector $\textbf{b}$ from vector $\textbf{a}$ by subtracting the respective components.

If $\textbf{a} = a_xi + a_yj$ and $\textbf{b} = b_xi + b_yj$

then $\textbf{a} - \textbf{b} = (a_x - b_x)i + (a_y - b_y)j$

Vector Subtraction
Fig 11.4 - Vector Subtraction

Example 11.4: Given $\textbf{a}=4i+5j$ and $\textbf{b}=6i+3j$ find $\textbf{a}-\textbf{b}$.

$\textbf{a}$=$4i+5j$
$\textbf{b}$=$6i+3j$
$\textbf{a}-\textbf{b}$=$(4-6)i+(5-3)j$
 =$-2i+2j$

11.5 Vector Dot Product

Vector Dot Product
Fig 11.5 - Vector Dot Product

The vector dot product or scalar product multiplies vector $\textbf{a}$ by vector $\textbf{b}$ to give a scalar result.

$\textbf{a}\ . \textbf{b} = a_x . b_x + a_y . b_y $

Another way to calculate the vector dot product is

$\textbf{a}\ . \textbf{b} = |a|\ . |b|\ cos(\theta) $

Example 11.5a: Given $\textbf{a}=2i-3j$ and $\textbf{b}=i+4j$ find $\textbf{a}\ .\textbf{b}$

$\textbf{a}$=$2i-3j$
$\textbf{b}$=$i+4j$
$\textbf{a}\ .\textbf{b}$=$2\ . 1 + (-3)\ . 4$
 =$-10$

Example 11.5b: Given $\textbf{a}=(3, 10^{\circ})$ and $\textbf{b}=(4, 70^{\circ})$ find $\textbf{a}\ .\textbf{b}$

$\textbf{a}$=$(3, 10^{\circ})$
$\textbf{b}$=$(4, 70^{\circ})$
$\textbf{a}\ . \textbf{b}$=$3\ . 4\ . cos(60)$
 =$6$

In addition to the vector dot product there is a vector cross product. The vector cross product of $\textbf{a}$ and $\textbf{b}$ produces a vector that is perpendicular to the plane containing $\textbf{a}$ and $\textbf{b}$. See section 11.10

11.6 3D Vectors

So far we have been looking at 2D vectors. To extend vectors to three dimensions we need to add an additional axis, the $z$-axis, which is perpendicular to both the $x$ and $y$ axes and, in this document, is pointing out of the page. The $z$-axis has a unit vector $j$ so the 3D vector $\textbf{a}$ may be represented as $\textbf{a} = a_xi + a_yj + a_zk$

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Fig 11.6 - 3D Axes

11.7 3D Vector Addition

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Fig 11.7 - Vector Addition

Adding 3D vectors is similar to adding 2D vectors. All we have to do is add the components for the $z$ dimension.

$\textbf{a} = a_xi + a_yj + a_zk$ and $\textbf{b} = b_xi + b_yj + b_zk$

so

$\textbf{a} + \textbf{b}$$=$$(a_x + b_x)i$
  $ + (a_y + b_y)j$
  $ + (a_z + b_z)k$

Example 11.7: Given $\textbf{a}=6i+5j-4k$ and $\textbf{b}=-10i+5j-5k$ find $\textbf{a}+\textbf{b}$.

$\textbf{a}$=$6i+5j-4k$
$\textbf{b}$=$-10i+5j-5k$
$\textbf{a}+\textbf{b}$=$(6-10)i+(5+5)j+(-4-5)k$
 =$-4i+10j-9k$

See figure 11.7

11.8 3D Vector Subtraction

We can subtract 3D vector $\textbf{b}$ from 3D vector $\textbf{a}$ by subtracting the respective 3D components.

If $\textbf{a} = a_xi + a_yj + a_zk$ and $\textbf{b} = b_xi + b_yj + b_zk$

then $\textbf{a} - \textbf{b} = (a_x - b_x)i + (a_y - b_y)j + (a_z - b_z)k$

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Fig 11.8 - Vector Subtraction

Example 11.8: Given $\textbf{a}=6i+8j+5k$ and $\textbf{b}=5i+14j-5k$ find $\textbf{a}-\textbf{b}$.

$\textbf{a}$=$6i+8j+5k$
$\textbf{b}$=$5i+14j-5k$
$\textbf{a}-\textbf{b}$=$(6-5)i+(8-14)j+(4-(-5))k$
 =$i-6j+10k$

See figure 11.8

11.9 3D Vector Dot Product

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Fig 11.9 - Vector Dot Product

The 3D vector dot product or scalar product multiplies vector $\textbf{a}$ by vector $\textbf{b}$ to give a scalar result.

$\textbf{a}\ . \textbf{b} = a_x\ . b_x + a_y\ . b_y + a_z\ . b_z $

Another way to calculate the vector dot product is

$\textbf{a}\ . \textbf{b} = |a|\ . |b|\ cos(\theta) $

In 3D $|a|= \sqrt{a_x^2 + a_y^2 + a_z^2}$ and $|b|= \sqrt{b_x^2 + b_y^2 + b_z^2}$

Example 11.9a: Given $\textbf{a}=-4i+6j+5k$ and $\textbf{b}=5i+8j+4k$ find $\textbf{a}\ .\textbf{b}$

$\textbf{a}$=$-4i+6j+5k$
$\textbf{b}$=$5i+8j+4k$
$\textbf{a}\ . \textbf{b}$=$-4\ . 5 + 6\ . 8 + 5\ . 4$
 =$48$

Example 11.9b: Find the angle between the two vectors from Example 11.9a.

We know $\textbf{a}\ .\textbf{b}$=$a_x\ . b_x + a_y\ . b_y + a_z\ . b_z$
We also know $\textbf{a}\ .\textbf{b}$=$|a|\ . |b|\ cos(\theta) $
So we can write
$a_x\ . b_x + a_y\ . b_y + a_z\ . b_z$=$|a|\ . |b|\ cos(\theta) $
rearranging $cos(\theta)$=$ \frac{a_x\ . b_x + a_y\ . b_y + a_z\ . b_z}{|a|\ . |b|} $
which means $\theta$=$ cos^{-1}(\frac{a_x\ . b_x + a_y\ . b_y + a_z\ . b_z$}{|a|\ . |b|}) $
$|a|$=$ \sqrt{a_x^2 + a_y^2 + a_z^2}$
 =$ \sqrt{(-4)^2 + 6^2 + 5^2}$
 =$ \sqrt{77}$
$|b|$=$ \sqrt{b_x^2 + b_y^2 + b_z^2}$
 =$ \sqrt{5^2 + 8^2 + 4^2}$
 =$ \sqrt{105}$
$\theta$=$ cos^{-1}(\frac{-4\ . 5 + 6\ . 8 + 5\ . 4$}{\sqrt{77}\ . \sqrt{105}}) $
 =$ cos^{-1}(\frac{48}{89.92}) $
 =$ 1.01 $ rad or $57.74^{\circ}$

11.10 Vector Cross Product

The vector cross product is the product of two vectors $\textbf{a} \times \textbf{b}$ that results in a vector. The resultant vector is perpendicular to the plane containing $\textbf{a}$ and $\textbf{b}$ and is in the direction a right hand thread or corkscrew would travel when rotated from $\textbf{a}$ to $\textbf{b}$. This means $\textbf{a} \times \textbf{b}$ is on the same unit vector as $\textbf{b} \times \textbf{a}$ but points in the opposite direction.

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Fig 11.10 - Vector Cross Product

There are two ways to calculate the vector cross product:

Example 11.10: Given $\textbf{a}=-4i+6j+5k$ and $\textbf{b}=5i+8j-4k$ find $\textbf{a} \times \textbf{b}$.

$\textbf{a}$=$-4i+6j+5k$
$\textbf{b}$=$5i+8j+4k$
$\textbf{a} \times \textbf{b}$=$\begin{vmatrix} i & j & k \\ -4 & 6 & 5 \\ 5 & 8 & 4 \notag \end{vmatrix} $
 =$i(24-40)$
  $-j(-16-25)$
  $+k(-32-30)$
 =$-16i+41j-62k$

See figure 11.10

11.10 3D Direction Cosines

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Fig 11.10 - Direction Cosines

An alternative way to specify a vector is to give its magnitude and the angle it makes with each of the axes. The three angles are usually called $\alpha, \beta$ and $\gamma$ (alpha, beta and gamma) where $\alpha$ is the angle between the vector and the $x$-axis, $\beta$ is the angle between the vector and the $y$-axis and $\gamma$ is the angle between the vector and the $z$-axis.

The cosines are often called $l$, $m$ and $n$ where $l=cos(\alpha)$, $m=cos(\beta)$ and $n=cos(\gamma)$

The magnitude of vector $\textbf{a}$ is given by $|a| = \sqrt{a_x^2+a_y^2+a_z^2} $ so:
$l$$\ =\ $$cos(\alpha)$$\ =\ $$a_x/|a|$
$m$$\ =\ $$cos(\beta)$$\ =\ $$a_y/|a|$
$n$$\ =\ $$cos(\gamma)$$\ =\ $$a_z/|a|$

Example 11.10: Given $\textbf{a}=4i+6j+5k$ find the direction cosines.

$\textbf{a}$=$4i+6j+5k$
$| \textbf{a}|$=$\sqrt{16 + 36 + 25}$
  $8.77$
$cos \alpha$=$4/8.77=0.46\ rad$
$cos \beta$=$6/8.77=0.68\ rad$
$cos \gamma$=$5/8.77=0.57\ rad$
 
So $4i+6j+5k$=$[8.77,0.46,0.68,0.57]$

See figure 11.10

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