Vectors And Their Applications

Reflection 8 would go here

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 \(<-2, 5>\)

\[\text{Dot : } 6 * -2 + -4 * 5 = -32\]\[\text{Magnitude of vector 1: } \sqrt{6^2 + -4^2} = \sqrt{52}\]\[\text{Magnitude of vector 2: } \sqrt{-2^2 + 5^2} = \sqrt{29}\]\[\cos^{-1}({-32\over\sqrt{52}*\sqrt{29}}) = 145.491^\circ\]