Vectors

Definition of a Vector

Many of the quantities we will be dealing with such as velocity, acceleration and force have not only a magnitude or size but also a direction. Because of this, it is useful to use what we call vectors to describe these quantities. The graphical representation of a vector is a directed line segment.

Vector.jpg

First, we will look at vectors from a graphical perspective and then from a numerical perspective.

Graphical Representation

Vector Addition

To add two vectors place the second vector with its initial point coinciding with the terminal point of the first vector. The diagram below illustrates this process.

VectorAddition.jpg

A vector only has a direction and a magnitude associated with it and not a location. So when we move our vectors in order to line up the terminal and initial points we do not change those properties of the vector.
Below we see that the operation of vector addition is commutative. That is to say $\vec{u}+\vec{v}=\vec{v}+\vec{u}$

VectorAdditionParallelogram.jpg

Scalar Multiplication

We will see that there are several different types of multiplication defined on vectors. First, we will look at the multiplication of a vector by a real number or scalar. This is called scalar multiplication. Multiplying a vector by a scalar, $k$, stretches the vector by a factor of $k$. If $k<0$, it also flips the vector around to the opposite direction.

VectorScalarMult.jpg

Vector Subtraction

Now that we have a definition for addition and scalar multiplication we can define subtraction simply as the addition of the opposite. In other words $\vec{u}-\vec{v}=\vec{u}+(-1\vec{v})$.

VectSub.png

Numerical Representation

Component Form

In order to arrive at a numerical representation of a vector, we need to define what are called basis vectors. Think of these as the building blocks of all other vectors. The most natural thing to do is choose vectors that align with our three axes in the standard Cartesian coordinate system. In order to make things easy, we will make these basis vectors all have length one. In other words, they will be unit vectors. Now we just need to give these vectors names, so that we can refer to them. The three unit vectors, pointing in the the positive x,y, and z directions respectively are $\vec{i}$,$\vec{j}$, and $\vec{k}$. We call these the standard unit normal vectors

StandardUnitNormals.png

With these vectors along with the operations of addition and scalar multiplication, we can create any vector we want. For example, a vector that runs 1 unit in the positive x-direction, 2 units in the positive y-direction, and 3 units in the positive z-direction would be equivalent to the sum $1\vec{i}+2\vec{j}+3\vec{k}$.

SumOfNormals.PNG

For the purposes of vector arithmetic, the scalars in front of our standard unit normals are all we need. Therefore, as a shorthand, we will write vectors in what is called component form. The component form of a vector is given by

(1)
\begin{align} \vec{v}=<v_x,v_y,v_z> \end{align}

In our previous example we would have $\left < 1,2,3 \right >$
The quantities $v_x, v_y, v_z$ indicate the change in x, y and z respectively as you travel along the vector from its initial point to its terminal point.

Vectors consist of a magnitude and direction, but not a position. Because of this, when we are drawing vectors in our Cartesian space, we are free to draw them in the location that makes the most sense. When a vector is placed with its initial point at the origin, we say it is in standard position. When a vector is in standard position it points to the point $\left ( v_x,v_y,v_z \right )$. For example, out previous vector $\vec{v} = \left <1,2,3 \right >$ in standard position point to the point $(1,2,3)$

VectorPoint.PNG

Exercise
A vector $\vec{v}$ when in standard position points to the point $(6,-8,-7)$. Express $\vec{v}$ in component form

Addition/Subtraction

To add or subtract two vectors in Cartesian form, we simply add or subtract componentwise as follows.

(2)
\begin{align} \vec{u} \pm \vec{v}=<u_x \pm v_x,u_y \pm v_y,u_z \pm v_z> \end{align}

Exercise
Find $\left < -5,8,12 \right > + \left < 2,-2,0 \right >$

Exercise
Find $\left < 1,4 \right > - \left < -8,9 \right >$

Scalar Multiplication

To multiply a vector by a scalar (real number) in Cartesian form, we simply distribute the scalar to each component of the vector

(3)
\begin{align} k\vec{v}=<kv_x,kv_y,kv_z> \end{align}

Exercise
Find the vector $\vec{v}$ that points in the direction of $\left < 1,4,-8 \right >$ but is half its length

Normalization

In certain application it can be useful scale a vector up or down so that maintains its direction, but its has unit length (i.e. $|\vec{v}|=1$). This process is called normalizing a vector. To obtain a vector $\vec{u}$ in the same direction as $\vec{v}$, but with $|\vec{v}|=1$, we do the following,

(4)
\begin{align} \hat{v}=\frac{1}{|\vec{v}|}\vec{v} \end{align}

The resulting vector $\hat{v}$ is also refered to as its unit direction

Exercise
Normalize the vector $\vec{v}= <2,-4,5>$


Note that any vector other than the zero vector can be expressed as a product of its magnitude and unit directoion.(5)
\begin{align} \vec{v} = |\vec{v}|\hat{v} \end{align}

Exercise
Express the vector $\vec{v}= <2,-4,5>$ as a product of its magnitude and unit direction

Polar Form

In 2D we have an alternative way of expressing vectors numerically called Polar Form. In polar form, we describe the vector by its magnitude, $|\vec{v}|$, and the angle, $\theta$, that it makes with the positive x-axis. The image below shows the relationship between the cartesian components and the polar angle and magnitude.

PolarForm.png

By right triangle trigonometry we can relate the Cartesian and Polar forms by the following equations,

(6)
\begin{eqnarray} v_x=|\vec{v}|\cos(\theta) \\ v_y=|\vec{v}|\sin(\theta)\\ |\vec{v}|=\sqrt{v_x^2+v_y^2}\\ \tan (\theta)=\frac{v_y}{v_x} \end{eqnarray}

Notice that the conversion equation relating $\theta$ to $v_x$ and $v_y$ is not written explicitly as $\theta = \tan ^{-1} \left ( \frac{v_y}{v_x} \right )$. Recall that our inverse trigonometric functions have limits to their ranges. Specifically in this case, $-\frac{\pi}{2} < \tan ^{-1} \left ( \frac{v_y}{v_x} \right ) < \frac{\pi}{2}$. Therefore, if the vector has $v_x<0$, we would need to compensate accordingly.

Exercise
Express the vector $\vec{v}= <-3,4>$ in polar form

Exercise
Express the vector $\vec{v}= 8@40^{\circ}$ in cartesian form

Addition/Subtraction

The easiest way to add or subtract two vectors in polar form is to convert them to Cartesian calculate the sum or difference and then convert back to polar.

Exercise
Compute $10@80^{\circ}+2@150^{\circ}$

Scalar Multiplication

Since scalar multiplication only effects the magnitude of the vector, scalar multiplication can be done without converting to Cartesian.

(7)
\begin{eqnarray} &k<|| \vec{v} , \theta>=< k || \vec{v} || , \theta> \mbox{ for } k \geq 0\\ &k<|| \vec{v} , \theta>=< k || \vec{v} || , \theta + \pi> \mbox{ for } k < 0 \end{eqnarray}

Exercise
Compute $5\left ( 4@280^{\circ} \right )$

Exercise
Compute $-\frac{1}{2} \left ( 6@100^{\circ} \right )$

Normalization

Since normalization is just a scaling of the vector's magnitude so that $||\vec{v}||=1$, we again can avoid any conversion to Cartesian form. Therefore the normalization $\vec{u}$ of a vector $\vec{v}=<||\vec{v}||,\theta>$ in polar form would be,

(8)
\begin{align} \vec{u}=<1,\theta> \end{align}

Magnitude, Heading and Pitch

In 3D, we can represent a vector by three quantities,

  1. (M)agnitude - The length of the vector
  2. (H)eading - The angle that the projection of the vector into the xy-plane makes with the positive x-axis
  3. (P)itch - The angle the vector makes with the xy-plane

Notice that the first two quantities are the same as polar form. We only needed to add the quantity pitch to determine the angle by which the vector is diverging from the xy-plane. The image below illustrates the three quantities.

MagHeadPitch.jpg

We have the following formulas for converting between Cartesian form and Magnitude-Heading-Pitch form,

(9)
\begin{align} v_x=\|v\| \cos P \cos H\\ v_y=\|v\| \cos P \sin H\\ v_z=\|v\| \sin P\\ \|v\|=\sqrt{v_x^2+v_y^2+v_z^2}\\ \sin P = \frac{v_z}{\|v\|}\\ \cos H = \frac{v_x}{\sqrt{v_x^2+v_y^2}} \end{align}

Exercise
Express the vector $\vec{v}= <3,-2,4>$ in terms of magnitude, heading, and pitch

Exercise
Find the cartesian form of the vector with $M=10$, $H=200^{\circ}$ and $P=-30^{\circ}$

Direction Angles

Another useful description of vectors in 3D is by their direction angles. The direction angles represent the angles that the vector makes with the positive x-axis, $\alpha$, positive y-axis, $\beta$, and positive z-axis, $\gamma$. A given set of $\alpha,\beta,\gamma$ does not describe a unique vector. They describe all vectors that point in the same direction. These angles are useful in describing matrix rotations about an arbitrary axis as we will see later in the course. When describing a rotation axis, The axis the direction is important, but not its length.
The following formulas can be used to calculate direction angles.

Exercise
Compute the direction angles for the vector $\vec{v} = \left < -8,0,2 \right >$

(10)
\begin{align} \cos \alpha =\frac{v_x}{|\vec{v}|} \mbox{ , } \cos \beta =\frac{v_y}{|\vec{v}|} \mbox{ , } \cos \gamma = \frac{v_z}{|\vec{v}|} \end{align}
euler.jpg

One imprtant identity that arises from the formulas for these direction angles is the following,

(11)
\begin{align} ((\cos \alpha)^2 + ( \cos \beta )^2 + (\cos \gamma)^2 =\frac{(v_x)^2}{|\vec{v}|^2} + \frac{(v_y)^2}{|\vec{v}|^2}+ \frac{(v_z)^2}{|\vec{v}|^2} = \frac{v_x^2+v_y^2+v_z^2}{|v|^2}=\frac{|\vec{v}|^2}{|\vec{v}|^2}=1 \end{align}

Exercise
Compute the direction angles for the vector $\vec{v} = \left < -8,0,2 \right >$

Exercises

  1. Convert the Cartesian vector, $\vec{v}=<-1,2>$ to polar form
  2. Convert the polar vector $\vec{v}=5@220^{\circ}$ to Cartesian form
  3. Given $\vec{u}=<1,5,-8>$ and $\vec{v}=<-2,0,6>$, find $2\vec{u}-3\vec{v}$
  4. Add the vectors 4 cm @ 153º and 7 cm @ 25º.
  5. Starting at 17 feet east of the origin, an object moves 28 feet east, then 13 feet west, then 5 feet east. Find its new position, the total distance moved, and the total displacement.
  6. Find the magnitude and unit direction of the displacement vector $15\hat{x}-4\hat{y}$ km.
  7. Convert the velocity vector $<-9,4>$m/s to polar form.
  8. Convert the acceleration vector 16 $m/s^2$ @ 233º to Cartesian form.
  9. Find the angles that the force vector $31N\hat{x}-15N\hat{y}+22N\hat{z}$ makes with each of the three positive axes. ($N$ is for newtons, a unit of force.)
  10. A plane is heading along the vector $<25, 18, 4>$m. Determine its trig. heading and pitch, with respect to the world.
  11. A plane is traveling at 125 mph on a heading of 134º, with pitch of 7º . Determine the velocity vector in rectangular coordinates.
  12. A 3D acceleration vector of magnitude 41 m/s2 makes an angle of 56º with the positive x-axis, and 52º with the positive y-axis. Find the two possible sets of rectangular coordinates of this acceleration vector.
  13. Normalize the momentum vector $<-7,-4,-9>\frac{\mbox{kg}\cdot \mbox{m}}{\mbox{s}}$.
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