Math, Trigonometry and Vectors

Geometry

What is the angle $\beta$ equal to?

Trigonometry

Trig Definitions

Here's a familiar image.

This one might be a little less familiar, but the same rules apply.

Inverse functions

We can also use the inverse trigonmetric functions.

It's 450 meters from the corner of Hamilton Place and Broadway to 142nd St and Broadway. It's 489 meters from the same corner to 142nd if you walk along Hamilton Place. a) What is the angle between Hamilton and Broadway?

Pythagorean Theorem

We'll use this relationship all the time.

$$a^2 + b^2 = c^2$$.

It's 450 meters from the corner of Hamilton Place and Broadway to 142nd St and Broadway. It's 489 meters from the same corner to 142nd if you walk along Hamilton Place. b) How far is it from Broadway to Hamilton Pl. walking along 142nd St?

map

Scalars and Vectors

These are two different mathematical or physical entities.

Scalars: A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. (e.g. $20)

Vectors: A vector quantity is completely described by a number and appropriate units plus a direction. (e.g. person walks 2 km E)

Some Examples:

Scalars: Temperature, Speed, Distance, length, density

Vectors: Displacement, Velocity, Force, Weight

When thinking out the physical world, you should intuitively notice that certain quantities or phenomena have different effects depending on which way they are pointed, in other words, their effects depend on their direction. For example, it's a lot easier to walk with the wind blowing in the same direction as your motion, rather than the other way: walking against the wind. There are two vector quantities at play in this example. Your direction of motion (that would be inferred from your velocity vector) and the velocity of the wind. When they point in the same direction, your motion is aided by the wind, when they are in opposite directions, your motion is impeded. Not only do the magnitudes of these two quantities matter, but so do their directions. And thus, we need to use vectors.

Vector vs. Scalar example

A particle travels from A to B along the path shown by the dotted red line. This is the distance traveled and is a scalar

The displacement (change in position) is the solid line from (a) to (b). The displacement is independent of the path taken between the two points displacement is a vector (it has length and direction).

Notation

When writing math by hand, just put an arrow on top of the variable. This will indicate it is a vector: $\overrightarrow{A}$

In printed text, you'll see vectors either in bold face: $\mathbf{A}$ or with an arrow: $\overrightarrow{A}$

If we want to refer to the magnitude only of a vector quantity, we can use absolute value bars: $|A|$, or just in italics: $A$.

Wind Map

Properties of vectors

Two vectors are equal if they have the same magnitude and the same direction

$\mathbf{A} = \mathbf{B}$ if $|A| = |B|$ and they point along parallel lines

All of the vectors shown are equal in magnitude and direction, thus they are equal.

Addition of Vectors

Adding two scalar quantities is easy. We just add them like we would add any normal quantity.

However, vectors involve more math. We have to also take into account which way they are pointing.

Imagine we walk along two displacement vectors $\overrightarrow{A}$ and $\overrightarrow{B}$. What is the resultant displacement? Or, what is $\overrightarrow{A} + \overrightarrow{B}$?

Vector Addition: Graphically

To add $\overrightarrow{A} + \overrightarrow{B}$

Arrange the vectors tip to tail.
Connect the tip of $\overrightarrow{A}$ to the origin of $\overrightarrow{B}$.

Link to Vector Addition Sim.

Vector Addition: Sim

The two blue vectors are added together resulting in the black-dashed vector. You can drag each of the blue ones around, change its magnitude and direction, and observe the change in the resultant.

Which of the following is the resultant for $\overrightarrow{P} + \overrightarrow{Q}$?

Negative of a Vector

The negative of a vector is simply a vector with the same magnitude, but pointed in the opposite direction.

The resultant of $\overrightarrow{A} + (-\overrightarrow{A}) = 0$

Vector Subtraction

To Subtract two vectors, say $\overrightarrow{A} - \overrightarrow{B}$, all we need to do is add the negative of $\overrightarrow{B}$ to $\overrightarrow{A}$.

Since, $\overrightarrow{A} - \overrightarrow{B} = \overrightarrow{A} + (-\overrightarrow{B})$

Which of the following is the resultant for $\overrightarrow{P} - \overrightarrow{Q}$?

Multiplication of a Vector times a Scalar

The result of the multiplication or division by a scalar is a vector.

The magnitude of the vector is multiplied or divided by the scalar.

Vector Components

The components of a vector are the parts of a vector that point along a given axis. We'll use the Cartesian Coordinate System most often.

Here we see the $x$ and $y$ components of the vector $\mathbf{A}$.

We can see that the $x$ component, $\mathbf{A}_x$, points all along the $x$ axis, while the $y$ component, $\mathbf{A}_y$, points only along the $y$ axis.

Here's another vector $\mathbf{A}$ decomposed into its $x$ and $y$ components.

Here is a vector and its x and y components. Change the vectors magnitude and direction and observe the components.

Here are the components of $\mathbf{R}$:

$$R_x = +4, R_y = +3$$

Which diagram represents $\mathbf{R}$?

We can use these vector components to add two arbitrary vectors together. (notice that $\mathbf{A}$ and $\mathbf{B}$ are not at right angles to each other.)

We'll combine the components of $\mathbf{A}$ and $\mathbf{B}$ to get the components of $\mathbf{C}$.

$$\mathbf{C}_x = \mathbf{A}_x+\mathbf{B}_x$$ $$\mathbf{C}_y = \mathbf{A}_y+\mathbf{B}_y$$

Once we have the components of $\mathbf{C}$, we can use the pythagorean theorem to get the magnitude of $\mathbf{C}$.

$$|C| = \sqrt{C_x^2+C_y^2}$$

A car travels 20 km due N and then 35 km in a direction 60º W of N. Find the magnitude and direction of the car’s resultant displacement.

Unit vectors

A unit vector is a vector that has a magnitude of exactly 1, and points in a given direction.

Function of Unit vectors

Rather than always using the $\theta$ and magnitude of a vector to describe it, we can use the unit vectors.

$\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$

$\mathbf{A} = 5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}}$

Dot Product

There are two ways of multiplying vectors:

1. The dot product (or scalar product)

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

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

$\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z$

Cross Product

The second method produces another vector:

2. The cross product (or vector product)

$|\mathbf{a} \times \mathbf{b}| = a b \sin \phi$

This produces a third vector that points perpendicular to both the original vectors.

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

Graphs & Plots

Visual Information is Powerful.

Functional Relations

Constant
Linear

Quadratic

What is the sum of vectors $\mathbf{A} + \mathbf{B} + \mathbf{C} +\mathbf{D}$?

$\mathbf{R}$
$\mathbf{-R}$
$\mathbf{G}$
$\mathbf{-G}$
0