# Math Refresher — Vectors and Trig

## Vectors

A vector is something that has

*magnitude*(length, size) and

*direction*(left-right, north-south, etc.). First, we need to explain a special type of vector, called a

*unit*vector. It has magnitude and direction, but it’s special in that its magnitude (length) = 1. It’s called a unit vector because unit means “one.”

In Chapter 4, we described motion in 2 dimensions, \(x\) and \(y.\) Now, we’re going to learn how to describe and calculate in 3 dimensions, \(x,\) \(y,\) and \(z.\) In a Cartesian Coordinate system (\(x,\ y,\ z\)) we have a *convention* (i.e., standard notation) that the unit vector for the \(x\) direction is \(\hat\imath,\) for the \(y\) direction is \(\hat\jmath,\) and for the \(z\) direction is \(\hat k.\) The “hat” symbol over the letter means unit vector, or vector with magnitude of one.

So how do we use them? Say we have a vector, \(\bf r,\) that points from the origin to the (\(x, y, z\)) point of (2, 4, 5) meters. We write this as

$$ {\bf r} = 2\,\hat\imath + 4\,\hat\jmath + 5\,\hat k \; {\rm m}$$

writing the unit (m) at the end. The key thing in dealing with vectors is keeping the components (directions) separate.

### Adding Vectors

So how do I add (or subtract) two vectors? Say we have the above vector, \(\bf r,\) and to that we need to add a vector, \(\bf s,\) defined as

$$ {\bf s} = 4\,\hat\imath + 1\,\hat\jmath + 3\,\hat k \; {\rm m}$$

So we add each component separately,

$$ {\bf r} + {\bf s} = (2 + 4)\,\hat\imath + (4 + 1)\,\hat\jmath + (5 + 3)\,\hat k \; {\rm m}$$

which gives

$$ {\bf r} + {\bf s} = 6\,\hat\imath + 5\,\hat\jmath + 8\,\hat k \; {\rm m}$$

The sum of two vectors is a vector, and is called the

*resultant.*

### Subtracting Vectors

We can also subtract two vectors, in just the way you would expect. So the difference of these two vectors would be

$$ {\bf r} – {\bf s} = (2 – 4)\,\hat\imath + (4 – 1)\,\hat\jmath + (5 – 3)\,\hat k \; {\rm m}$$

which gives

$$ {\bf r} – {\bf s} = -2\,\hat\imath + 3\,\hat\jmath + 2\,\hat k \; {\rm m}$$

This is also called a

*resultant.*

### Adding Vectors Graphically

Say we had two vectors, \(\bf A\) and \(\bf B.\) We can add them

*graphically*by drawing them

*head-to-tail.*Then, the resultant, \(\bf C,\) is drawn from the beginning (tail) of \(\bf A\) to the head (point) of \(\bf B,\) like so

### Vector Magnitude

Continuing to use the \(\bf r\) and \(\bf s\) vectors that we defined above, the magnitude for \(\bf r\) is written like this \(\bf | r |\) or is written without the bold-face font, like so \({\bf| r |} = r.\) We can calculate the magnitude of a vector using the distance formula. For the vector \(\bf r\) above, its magnitude is written and calculated like so

$$ | {\bf r} | = \sqrt{x^2 + y^2 + z^2} = \sqrt{2^2 + 4^2 + 5^2}$$

which equals \(\sqrt{4 + 16 + 25} = 6.7\; {\rm m}.\)

### Multiplying Vectors — the Dot Product

There are two ways to multiply vectors. The first we’ll talk about is the

*dot product.*The dot product of two vectors is just a number (also called a

*scalar*). It’s calculated by multiplying each component separately, then summing them all together. For the two vectors above, their dot product is shown here

$$ {\bf r} \cdot {\bf s} = (2 \cdot 4) + (4 \cdot 1) + (5 \cdot 3) = 8 + 4 + 15 = 27\,{\rm m} $$

Another way to calculate the dot product is the product of the magnitude of each vector times the cosine of the angle between them, like so

$$ {\bf r} \cdot {\bf s} = | {\bf r} | | {\bf s} | \cos\theta $$

This version is best to use when you are given a diagram showing the vectors and the angle between them.

### Multiplying Vectors — the Cross Product

The other way to multiply vectors is called the

*cross*product. The cross product of two vectors is a vector. If you’ve studied matrices and know how to calculate the determinant of a matrix, that’s how the cross product is calculated. If not, the formula for the determinant is given below as a written-out formula. It looks like this:

$$ {\bf r} \times {\bf s} = \begin{vmatrix}

\hat\imath & \hat\jmath & \hat k \\

r_x & r_y & r_z \\

s_x & s_y & s_z

\end{vmatrix}$$

which expands to

$$ (r_y s_z – s_y r_z)\,\hat\imath – (r_x s_z – s_x r_z)\,\hat\jmath + (r_x s_y – s_x r_y)\,\hat k $$

which for the above values of \(\bf r\) and \(\bf s,\) we get

$$ {\bf r} \times {\bf s} = \begin{vmatrix}

\hat\imath & \hat\jmath & \hat k \\

2 & 4 & 5 \\

4 & 1 & 3

\end{vmatrix}$$

which expands to

$$ (4\cdot 3 – 1 \cdot 5)\,\hat\imath – (2 \cdot 3 – 4\cdot 5)\,\hat\jmath + (2\cdot 1 – 4\cdot 4)\,\hat k \; {\rm m}$$

and simplifies to

$$ {\bf r} \times {\bf s} = 7\,\hat\imath + 14\,\hat\jmath – 14\,\hat k \; {\rm m}$$

Similarly, if you are given a diagram showing the vectors and the angle between them, an alternate way to calculate the cross product is to multiply the magnitude of each vector time the sine of the angle between them, like so

$$ {\bf r} \times {\bf s} = | {\bf r} | | {\bf s} | \sin\theta $$

Since this product is a vector, it needs a direction. The direction of the cross product is given by the right-hand rule. Here’s how: orient your right hand so that your fingers point in the direction of the first vector (\(\bf r\) in this case). Then move your right hand so that you can bend your fingers to point the direction of the second vector (\(\bf s\) in this case). Your right thumb is now pointing in the direction of the cross product.

## Trigonometry

### Trig Functions

Trigonometry is about using ratios of the sides of a right triangle. Let’s use this triangle to illustrate:

$$ \sin\theta = \frac{\rm opp}{\rm hyp} \qquad \qquad

\cos\theta = \frac{\rm adj}{\rm hyp} \qquad \qquad

\tan\theta = \frac{\rm opp}{\rm adj}

$$

### Trig Identities

All trig identities can be derived from the following two definitions. First

$$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$

and second

$$ \sin^2 \theta + \cos^2 \theta = 1$$

We can show that these are true. Use the definitions for sine and cosine

$$ \frac{\sin \theta}{\cos \theta} = \frac{ \ \frac{\rm opp}{\rm hyp} \ }{\frac{\rm adj}{\rm hyp}} $$

invert and multiply, then cancel the “hyp,” giving

$$ \frac{\rm opp}{\rm hyp} \times \frac{\rm hyp}{\rm adj} = \frac{\rm opp}{\rm adj}$$

which equals \(\tan \theta.\) For the second one,

$$ \sin^2 \theta + \cos^2 \theta = 1 $$

now, substitute in the definitions

$$\Big( \frac{\rm opp}{\rm hyp} \Big)^2 + \Big( \frac{\rm adj}{\rm hyp} \Big)^2 = 1$$

Then, multiply through by \({\rm hyp}^2,\) giving

$$ {\rm opp}^2 + {\rm adj}^2 = {\rm hyp}^2 $$

which is the Pythagorean Theorem.

## Orders of Magnitude

This is typically used to compare two numbers. If the large number divided by the smaller number gives a quotient less than 10, then the two numbers are of the same order of magnitude. One order of magnitude is a factor of ten, two orders of magnitude is a factor of 100, and so on. For example, 47 and 96 are of the same order of magnitude. The number 1200 is 2 orders of magnitude greater than 12. Where does this come from? Like so: \(1200 / 12 = 100;\) then \(100 = 10^2\) — so the exponent (power) of the ratio is the order of magnitude.

## Significant Figures

Significant figures are how many digits are left after removing all leading and trailing zeroes. For example, 24.63000 is 4 significant figures (remove the 3 trailing zeroes and count the digits). A second example: 00324.40302400 has 9 significant figures (remove the leading and trailing zeroes and count the digits). This is useful when working a problem with a calculator and knowing how many figures to keep at each step of the calculation.

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