Unit 3: Vectors
By Kieran Dias-Lalcaca and Ryan Ngo
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Vectors: A refresher course
Vectors: A refresher course
Table of Contents
Table of Contents
The year is 2025. You are either recently graduated from college, or an upcoming senior in college. You are at your first internship. Your boss comes up to you and says, "You have a test on vectors tomorrow. If you fail you're fired."
What do you do in this very unlikely situation?
Just refer to that PreCalculus Honors Class you took at the Haverford School with Mr. Bridge when you were either a sophomore or junior. Luckily, you stumble across your notes for the course, which are unfortunately collecting dust.
Your table of contents looks something like this:
What is a vector?
What is a vector?
A vector is defined as a quantity in mathematics that holds both a magnitude and direction. It is also referred to as a directed line segment that represents a quantity. Vectors are comprised of a series of scalars, each of which account for the direction the quantity travels, and how far it goes (magnitude).
A good visualization of a vector might be to think of an airplane; the pilot is instructed by the traffic control tower to fly a certain distance (magnitude), while maneuvering his plane in a certain way to get to the correct geographical destination (direction).
Together we will dive deeper into how we are able to calculate the valuable foundations to vectors, magnitude and direction, and the different topics we covered throughout the unit to fully understand vectors as a whole.
Important Terms:
Important Terms:
There are several terms that are essential to recognize and understand prior to diving deep into the investigation of what vectors are and what they do:
There are several terms that are essential to recognize and understand prior to diving deep into the investigation of what vectors are and what they do:
- Vector quantity - these are the quantities of vectors expressed on the basis of force, displacement, or velocity. These quantities will have both magnitude and direction.
- Scalar - the quantities expressed in vectors that only incorporate magnitude and not direction. Some good examples of these would be speed, time, or volume.
- Magnitude - otherwise known as the "absolute value" of the vector, the magnitude is its length.
- Unit vector - This a vector that is defined as the normed vector quantity and has the magnitude of 1. This effectively describes the three dimensional vectors i,j, and k in their own capacities.
- Position vector - a variation of a vector that begins at the origin of a coordinate plane and ends at a specific point. Based on its name, one would be able to derive what points the vector lies on by look at the coefficients in the vector equation.
- Displacement vector - This is a vector that is formed when determining the difference between a vector's starting and ending position.
- Resultant vector - This is the vector that is formed as a result of adding the two vectors together, essentially being the sum. This vector runs from the the head of the first vector to the tail of the second vector.
- Normal vector - vector that is perpendicular to the plane that a given vector lies on. To put into perspective, in 2D, the "normal" to a curve at a certain point is perpendicular to the tangent line to the curve at that point. When provided a standard equation of a vector, a(X-Xo) + b (Y-Yo) + c (Z-Zo), one should note that the coefficients (a, b, c) are potential orthogonal vectors to the plane.
Two dimensional Vectors
Two dimensional Vectors
To understand vectors in their full glory, we must first understand what they are in two dimensional space.
In two dimensional space each of the horizontal and vertical components are looked at as individual parts that allow for a vector to be formed. These parts are referred to as the horizontal(x axis) and vertical components(y axis).
This can also be represented by a magnitude, and an angle indicating direction.
Three Dimensional Vectors
Three Dimensional Vectors
Vectors are not only two dimensional, however. They are also able to be expressed in the third dimension. The standard 3D unit vectors include i, j, and k, which are all parallel to the x, y and z planes, respectively.
The diagram on the left gives a good example of vectors in a 3D space with cartesian coordinates.
Example Question:
Example Question:
Below is a great example of a problem that takes two 3D vectors in component form and asks for various operations, including the unit vector of a particular vector (a), similar to the top question. The second question simply asks the difference, and then the third question asks the sum of vector a and 7/10 of the combination of the two vectors, vector AB.
These problems are great examples of the complexity and versatility of vectors in 3D planes.
Here is a great example of calculating the Unit Vector of a 3D vector given the coordinates of a 3D vector. The coordinates are squared and then added, and the square root of the sum is extracted. The unit vector is then each coordinate being the numerator with the newly calculated square root of the sum of all three squared coordinates being the denominator.
Example Problems for adding/subtracting vectors:
Example Problems for adding/subtracting vectors:
Examples of this are below, with 3D vectors expressed in their components and the calculations for the differences or sums of them are very straightforward.
Adding and subtracting vectors
Adding and subtracting vectors
Basic arithmetic involving two vectors is very simple, and can be used When provided two vectors in interval notation, no matter what dimension the vector is expressed in. Simply, if we intend to add or subtract the two vectors together, then we would add or subtract the corresponding variables in each vector.
Another way to add or subtract vectors is through the visualization of the vectors by drawing them. One can add the vectors by connecting them from head to tail and measuring them from the bottom of the first vector to the top of the last. Subtracting would be the opposite, utilizing an "opposite" resultant vector to measure the difference.
Dot Product
Dot Product
Example:
Example:
Below are great examples of the basic use of the dot product when provided two 2D vectors in component form.
The relationships between the two are expressed below and the different variations of vector a (dot) vector b are expressed in either multiplying corresponding x and y values of both vectors by each other and adding them. T
he other variation is taking the absolute values of each vector and multiplying them by the cosine of the angle that is formed when the two vectors are connected.
While the dot product was not necessarily a subunit covered, it was a foundational concept that we had to build upon in order to expand to more complex real-world applications further on in the unit.
The dot product, in a computational sense, is the product of magnitudes of two vectors and the cosine of the angle between them.
Therefore, the dot product, allows us to find the length of the vector or the angle in between two vectors when given one or the other.
As you can see, the equation for the dot product is |a| (absolute value of vector a) * |b|(absolute value of vector b) * cos theta (angle created by two vectors).
Cross Products
Cross Products
The cross product is another computational variation of the dot product. It is essentially the product of the magnitudes of two vectors and the sine of the angle between them.
Unlike the dot product, cross product would also allow us to find a vector that is perpendicular to the plane that is covered by two vectors.
This is called the normal vector.
The normal vector is orthogonal to the plane which is formed by the other two vectors.
Then you can now put the cross product to use, and when calculating the coefficients for i, j, or, k you should cross multiply the bottom and top values of the OTHER variables (when solving for i you multiply value 5j by 8k to get 40i and then the 7j by 4k to get 28i.
You subtract the two variables to get the new coefficient, which is 12i, and repeat the process until you calculate for a new vector. You then set up a distribution style equation, and use one of the points as the coefficient on the outside of the parenthesis.
Inside the parenthesis you then plug in the A, B, and C variables and subtract by the coefficients of the new vectors. You then the equation to zero, indicating it is a system of equations. You will then proceed to distribute, simplify, and eventually derive the original system of equations for these points.
Cross Product Examples:
Cross Product Examples:
Pictured on the left is an example of the cross product being used in the context of a three sided figure in a coordinate plane.
Here, you are given three points and their coordinates on that three dimensional plane. We are then asked to solve for the points standard form, which is expressed as a 3-variable system of equations.
To your luck, you are also provided of two vectors formed by the shape in the plane, vector AB and AC, in "i, j, k" notation, the points in parenthesis showing the distance between them and then simplified.
Planes In Space
Planes In Space
One is able to create a plane or determine an equation of a plane when provided points that lie on the plane. One can do this by following a multi-step process:
- Determine the displacement vectors between the given coordinates. As you reviewed earlier, this is done through basic arithmetic, by finding the difference between the corresponding x, y and z points of specific planes.
- Locate the normal vector perpendicular to the plane in which the vectors and its coordinates lie. As we learned earlier in the terminology section, the normal vector is able to be determined by simply identifying the coefficients when provided the equation of a vector in standard form. When given the points and not the equation, one must utilize the cross product, another topic you also learned (remember how units built on skills back then??) to determine the coefficients and ultimately the normal vector.
- With the normal vector identified, revert the variables to a proper 3D vector notation, (i, j, and k form) and then derive the equation of the plane. The only thing left to do is take one of the points and plug it back into the newly derived equation of the plane in order to determine the other side of the plane.
- Combine all variables on both sides of the equation! You now have the equation of a plane in space.
Example:
Example:
Law of Cosines
Law of Cosines
The first subtopic that was addressed in this unit was the law of cosines, which was essentially an expansion of a vector identity.
This mathematical concept would allow us to calculate the magnitude of a certain quantity in space, specifically given the angle (direction) it is moving in.
As stated on the diagram on the left, the incorporation of each angle in the triangle allows for mathematicians to try and solve for the length (magnitude) of side when given other variables, such as the magnitude of the other vectors in the plane and the angles they make when they converge.
The equations attached will provide clarity on its interchangeability --one can utilize them to solve for missing angles or magnitudes of vectors. The angles formed by two vectors will also be instrumental in determining the direction that the vectors are pointed in.
Example Problem:
Example Problem:
This question is a great real world example of how to add two vectors that are not aligned head to tail, utilizing velocity vectors.
The first step would to be to try and draw a diagram that represents the ship's movement through a vector that is 9 units long, as well as a vector that represents the current's movement, which would be 4 units long.
Align these two vectors tail to tail to visualize the movement of the ship against the current at an angle of 130 degrees. Then, move one of the vectors so that they are then head to tail. This translation would cause the formation of a resultant vector. Draw the resultant vector from the head of one vector to the tail of the other.
As you can see by the diagram on the right, the 4 unit long vector (4 knots) is parallel to its original position. Therefore, the 9 unit vector (9 knot) is known as a transversal, or a line that passes through two lines on the same plane at two distinct points. Here, the 9 unit vector cuts through the now parallel original and new 4 unit vector. The angle formed between the resultant and original vectors is supplementary to the original 130 degrees in which the ship is traveling (50 degrees).
The law of cosines then comes into play in solving for the resultant vector and the new angle of the ship in relation to the shore formed as a result of the current (above) ^
Area of a Triangle and the Law of Sines
Area of a Triangle and the Law of Sines
Using the Law of Sines:
Using the Law of Sines:
The Law of Sines, which evaluates the relationship between lengths of the sides in a triangle in relation to sine of the angles those sides form, can be used to calculate the area of a triangle by utilizing the formulae on the left. The formulae are derived from the original triangle area formula, but takes into account the length of two sides and the sine of the angle of the other, which could be helpful in determining the length (magnitude) of specific vectors or the angles created (direction) when given the area.
Example:
Example:
In this basic example, we are tasked to determine the area of the triangle given the length of all 3 sides and the angle formed by 2 of them.
The application is simple:
Ambiguous Case
Ambiguous Case
Having went over the law of sines and cosines, we then moved onto ambiguous case, which essentially states that the law of sines can be used to determine a missing side or measure of a triangle when given two sides, and the angle opposite to one of those angles (Side-Side-Angle) (SSA).
It also allows us to determine how many triangles can be formed, or whether or not one can be formed, on the basis of specific measures meeting a certain measurement criteria.
Basic Example Problem:
Basic Example Problem:
In this basic question, you are tasked with finding the measure of a missing side in a triangle given two sides and an angle formed by the two sides provided. You should be able to recognize that law of cosines is able to be utilized to solve for the missing side, and that a rearrangement of the equation should lead you to the right answer.
Here is a more advanced application of the rules of the ambiguous case, where you are provided the distance from Ocean City to a radio station, as well as the range that the radio station is able to reach. You are also given the angle in which Ocean City is oriented in relation to the radio station. Here, your inductive reasoning skills that you learned from the ambiguous case are put to the test in determining how you are to go about solving for the distance between OC and the farthest east that people can hear the station.
Advanced Example Problem:
Advanced Example Problem:
Direction Cosines and Angles
Direction Cosines and Angles
In this unit, we learned about how one can find the angle of a beam that extends in space from the origin.This may seem like a complicated process but is really quite easy to accomplish. The Direction Angle is simply the angle of rotation that a beam has from its respective axis. This can be signified by the greek letters alpha, beta, and gamma.
Here we can see the definition for direction angles.
A direction angle is simply using the relationship of triangle sides and angles, through trigonomentry, to represent, and mathmatically find, what a certain angle is.
Here we can see how direction angeles and their relation the the Pythagorean therom. It is with this relation that the diretion cosines can give a unit vector. This is a useful application, and tool that allows for another way to find out information
Example Problem:
Example Problem:
Find the direction cosines and direction angles of the positon vector to the point of: (2,-5,3)
Vector equations for lines in space
Vector equations for lines in space
Vector equations for lines in space work in an interesting way. Due to the complicated nature of their forming, they work as the displacement vectors form the origin, in unit vector form.
Here we can see how the general equation works with both the idea that using two separate position vectors and the unit vector, and then making a displacement vector, the line can be formed.
It is with this process that any line can be mapped in 3-D space.
Example problem:
Example problem:
Find the vector equation for the line parallel to V containing the given point:
v= 2i-3j+4k
P=1,-8,-5
Thus, concludes the vital information we learned in Chapter 3 about vectors.
Thus, concludes the vital information we learned in Chapter 3 about vectors.
We hope our tutorial not only helped you save your internship by reporting vital information about vectors back to your boss, but really helped you relive some of your high school memories. Now get back to work you washed up alum!
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