Chapter 6 - Vectors
| Monday, 30 August 2021 | |
| 4-minute read | |
| 802 words | |
6.1 Law of Sines
\[ \sin \frac{A}{a} = \sin \frac{B}{b} = \sin \frac{C}{c} \]
The Law of Sines works best with AAS and ASA triangles.
It can work with SSA triangles. It may give you two solutions, one solution, or no solution at all.
Area of a Triangle
\[ A = \frac{1}{2}ab \sin C\]
Examples
Given \(a = 7\), \(b = 6,\) and \(A = 26.3^\circ\), find \(B\).
\(B = 22.320^\circ\)
Given \(a = 6\), \(b = 7\), and \(A = 30^\circ\), find \(B\).
\(B = 28.377^\circ\)
6.2 Law of Cosines
There are three formulations for the law of cosines:
\[ a^2 = b^2 + c^2 - 2bc\cos A \]
\[ b^2 = a^2 + c^2 - 2ac\cos B \]
\[ c^2 = a^2 + b^2 - 2ab\cos C \]
In a SSS triangle, try to solve for the biggest side for easier computations.
Heron's Formula
\[ \sqrt{s(s-a)(s-b)(s-c)} \textrm{where } s \textrm{ is the semiperimeter} \]
6.3 Vectors
Quantities can be divided into two formats: scalars and vectors. A scalar is a single numerical value which represents something such as temperature, time, or mass. A vector is something which contains a magnitude and a direction such as force or velocity.
\(\vec v\) - A vector
\(||v||\) - Magnitude of a vector
Important details for vectors
- Magnitude
- Direction
- Components
- Unit Vector Notation
Magnitude
The magnitude can be found using the Pythagorean Theorem.
$ a^2 + b^2 = c^2 $
$ ||v||^2 = ||v_x||^2 + ||v_y||^2 $
Direction
A vector's direction can be determined by angles or by compass bearings.
A \(0^\circ\) angle is always in the standard position, but a \(0^\circ\) compass bearing is always due north.
Components
\(\langle a, b \rangle\) or \(\langle \Delta x, \Delta y \rangle\)
Unit Vector Notation
\(v = xi + yj = \langle x, y \rangle\)
Vectors as Directed Line Segments (Examples)
- Consider the points \(A(2, 5)\) and \(B(4, -7)\).
If \(A\) were the initial point of a vector \(\vec u\) and \(B\) was its terminal point, how could we describe vector \(\vec u\)?
$⟨ 4, -7 ⟩ - ⟨ 2, 5 ⟩ = ⟨ 2, -12 ⟩
\(4 + 144 = ||v||^2\)
\(v = \sqrt{148}\)
- \(a = (-5, 7), b = (0, -3) c = 4i - 3j\)
\(\vec a + \vec c = \langle -5, 7 \rangle + \langle 4, -3 \rangle = \langle 1, 4 \rangle\)
- A plane flies 2.4 hours at 110 mph on a bearing of 40 degrees. The plane then turns and flies 1.8 hours at 130 mph on a bearing of 140 degrees. How far is the plane from its initial point?
Dot Product
Given two vectors such that \(\vec v = \langle v_x, v_y \rangle\) and \(\vec u = \langle u_x, u_y \rangle\)
\[ \vec u \cdot \vec v = u_x v_y + u_x v_y \]
When two vectors \(\vec v\) and \(\vec u\) are orthogonal, then \(\vec v \cdot \vec u = 0\)
The Angle Between Two Vectors
To find the angle between two vectors, the following formula is used:
\[ \cos \theta = \frac{\vec u \cdot \vec v}{||\vec u|| ||\vec v||} \]
Examples
- Find the angle between the vectors \(\vec u = \langle 4, 3 \rangle\) and \(\vec v = \langle 3, 5 \rangle\)
\(\theta = \cos^{-1}(\frac{12 + 15}{5\sqrt{34}})\)
\(\theta = 0.387\) or \(22.166^\circ\)
6.5 Complex Coordinates
Recall that complex numbers can be written in the format \(z = a + bi\), where \(a\) is the real component and \(b\) is the imaginary component. This is very similar to unit vector notation \(\vec v = ai + bj\).
A modulus of a complex number is the same as the magnitude of a vector.
An argument of a complex number is the same as an angle of the vector.
Example
Find the absolute value (modulus) of the complex number \(z = -2 + 5i\)
\[ z = \sqrt{-2^2 + 5^2} = \sqrt{29} \]
Find the argument of the complex number \(z = -2 + 5i\)
\[ z = 180^\circ - \tan^{-1}(\frac{5}{2}) = 111.801^\circ \]
Trigonometric Form
Using the modulus \(r\) and the argument \(\theta\), create another way of representing the complex number \(z\).
\[ z = r \cos \theta + i r \sin \theta \]
This can be shortened or abbreviated to
\[ r c i s \theta \]
Given only the components \(a\) and \(b\), you can use the formula
\[ z = \sqrt{a^2 + b^2} \cos(\tan^{-1}(\frac{b}{a})) + i \cdot \sqrt{a^2 + b^2} \sin(\tan^{-1}(\frac{b}{a})) \]
Moivre's Theorem
\[ (r \cis \theta^\circ)^x = r^x \cis x \theta^\circ \]
Examples
- Write the complex number \(z = -2 - 2\sqrt{3}\) in trigonometric form.
\[ 4 \cos 60^\circ + 4 \sin(60^\circ) i \]
- Find all of the fifth roots of \(-1\).
There are five possible answers since it is the fifth root.
Rewrite as a complex number
$$ z = -144