﻿ Vectors
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Vectors
Prerequisites
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Pythagorean Theorem
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Right Triangles
• A right triangle, or right-angled triangle, is a triangle in which one of the corners has an angle of 90o
• For example: • To show that it is a 90o angle we add a little square in the corner: This is also called a 'right angle'
• The longest side of the triangle is always the one opposite the right angle. This side is called the 'hypotinuse': Pythagorean theorem
• The pythagorean theorem can be used to find the length of the hypotinuse given the length of the other two sides.
• if a, b, and c are the length of the three sides: • The pythagorean theorem states that:
c2 = a2 + b2
Example
• if a = 4 and b = 3, calculate the length of c • Using the pythagorean theorem, we know that: • Now we just solve for c: • Finding other sides
• We can also use the pythagorean theorem to find the length of any side, given that we know the other two.
• For example: if a = 8 and c = 10, calculate the length of b • Using the pythagorean theorem, we know that: • Now we solve for b: • Code
import math

a = 4
b = 3
c = math.hypot(a, b)

print(c) # prints: 5


Vectors
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Definition
• Vectors are mathematical objects which represent change.
• They are the complement to coordinates, which represent fixed points.
• They have two properties:
• Direction
• Magnitude (or length)
• They are often written as how much change they apply in each dimension:
• A two dimensional vector that changes 3 in the first dimension and 5 in the second dimension is written as [3, 5]
• When writing a variable which holds a vector, an arrow is placed above its symbol: LaTeX: \vec a = [3, 5]
Vectors in one dimension
• A vector with only one dimension is written as a single value enclosed in a set of square brackets: • This can also be represented graphically: • Magnitude:
• The magnitude of a one dimensional vector is simply the absolute value of its only element.
• The vectors  and [-5] both have a magnitude of 5
• Direction:
• A one dimensional vector can only have two possible directions: forwards and backwards.
• The vectors  and [-5] have the same magnitude but different directions.
• The vectors [-3] and [-5] have different magnitudes but the same direction.
Vectors in two dimensions
• A vector with two dimensions is written as a pair of values enclosed in a set of square brackets: • This can also be represented graphically: • Magnitude:
• The magnitude of a two dimensional vector can be calculated using the Pythagorean theorem.
• The two components of the vector form two sides of a right triangle.
• The magnitude of the vector is equal to the hypotinuse.
• For example: • If we apply the Pythagorean theorem: Vectors in more than two dimensions
• Vectors can be defined in as many dimensions as desired.
• For example, a 5 dimensional vector would look like this: • Magnitude:
• To calculate the magnitude of a vector with n dimensions we use the following formula: • For example, the magnitude of: is Code
In python, vectors are represented as arrays.
import math
import numpy as np

vector = [3, 6, 2]

magnitude = math.sqrt(np.dot(vector, vector))
print(magnitude) # prints: 7.0

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