Vectors And Their Applications¶
Different forms of vectors¶
Source: Section 1.6 Example 2
Explanation:
This artifact demonstrates the different forms of vectors.
In the problem, the vector is in coordinate form.
I convert it to component form using the Head Minus Tail Rule, and then I convert it to magnitude form by taking its’ magnitude.
Artifact:
Find the magnitude of the vector \(v\) represented by \(\overrightarrow{PQ}\) where P = (-3, 4) and Q = (-5, 2)
\((2-4, -5-(-3)) = (-2, -2) \text{ Using the Head Minus Tail Rule}\\ |v| = \sqrt{(-2)^2 + (-2)^2} = 2\sqrt{2} \text { Component form to magnitude}\)
Vector application¶
Source: #2 from quiz 6.1 & 6.2
Explanation:
This artifact demonstrates vector application.
Artifact:
A boat is on a bearing of \(210^\circ\) traveling at 32 mph.
If it is in a 10 mph current that is on a bearing of \(273^\circ\), what is the boats ground speed and direction?
Vector for the boat: \(32<\cos{240}, \sin{240}> \text{ }=\text{ } <-16, -27.713>\)
Vector for the current: \(10<\cos{177}, \sin{177}> \text{ }=\text{ } <-9.85, 1.736>\)
Sum of the two vectors = \(<-25.986, -27.1894>\)
Speed = \(|<-25.986, -27.1894>| = \text{ 37.611 mph}\)
\(tan^{-1}({-27.1894 \over -25.85}) = 46.296\)
Bearing = \(90^\circ - 46.296^\circ = 43.5538^\circ\)
Finding the angle between two vectors¶
Source: #3 from quiz 6.1 & 6.2
Explanation:
This artifact demonstrates finding the angle between two vectors.
I found the answer to the problem using the following formulas:
- Angle between two vectors \(v\) and \(u\): \(\cos^{-1}({v*u \over |v| * |u|})\)
- \(<u_1, u_2> * <v_1, v_2> = u_1*v_1 + u_2*v_2\)
First I found the dot product of the two vectors, their I found their individual magnitudes.
Then, all I had to do was plug into the equation and solve for the angle.
Artifact:
Find the angle between the vectors \(<6, -4>\) and \(<-2, 5>\)