Vector Basics
Here we will learn about basics of vectors:[list] [*]Definitions [*]Properties [*]Operations [/list]
Table of Contents
- Equality and Examples
- Scalars or Vectors
- Vector Equality
- Vector Equality Practice
- Basic Vector Operations
- Vector Addition
- Vector Subtraction
- Scalar Multiplication
- Practice
- Vector Equation Game
- Finding Vectors Given Magnitude
- Component Form
- Vector Components
- Vector Component Addition
- Negative Vector's Components
- Scalar Multiplication Using Components
- Parallel Vectors
- Parallel Vectors
- Scalar Product
- Vector Projections
- Scalar Product
Scalars or Vectors
Below are 12 statements. Drag each statement into its correct category: scalar or vector.[br][br]If the answer is correct, a green check mark will appear.
Vector Addition
The applet below shows the addition of two vectors. You can move around the points, and then use the sliders to create the sum. For question 5, push "Combine Initial" to have the two vectors start at the same place.[br][br]Answer the questions, and move around the points to check various cases, until you are convinced of the property.
[b]Question 1: [/b]Explain in your own words, how to add vectors geometrically. [b]Note: [/b]be specific in your answer.[br][br][b]Question 2: [/b]For normal number addition, we have a special number "0", where [math]0+x=x+0=x[/math] . Can you create a [color=#ff0000]zero vector 0 [/color]such that [math]u+0=0+u=u[/math] for any vector [math]u[/math]? Describe the properties of this zero vector.[br][br][b]Question 3: [/b]For normal numbers [math]x[/math], we have a "negative number" [math]-x[/math] where [math]x+\left(-x\right)=\left(-x\right)+x=0[/math]. Choose any vector [math]u[/math] and then create its [color=#ff0000]negative [/color][math]-u[/math] (satisfying the property above). Repeat this for multiple vectors. Hence, describe how to find a negative of a vector.[br][br][b]Question 4:[/b] Consider two vectors [math]u[/math] and [math]v[/math]. Describe how to subtract the vectors (finding [math]u-v[/math]).[br][br][br]Since vectors can be freely moved, we will move them to start at the same position (to make it easier to visualize). To do this, push the "Combine Initial" button, and check "Show Both" to see both at the same time. Then, answer question 5.[br][br][b]Question 5: [/b]Is vector addition associative? [b]Hint:[/b] Associative means [math]u+v=v+u[/math]. Is this the same as real numbers?
Vector Equation Game
Type in the correct equation, linking the six vectors shown. [b]Note: USE == FOR EQUALITY, NOT =[br][br][/b]Once you hit enter, you will see if it is correct or not, and then push "Next Round", to go the next round. There are 10 rounds total, at the end, you will see your final score.
Vector Components
Below is vector u, with two red points you can drag. Push "Show Components" to see the components of the vector. Play around with the application, and then answer the questions below.
[b]Question 1: [/b]Explain in your own words, what a vector's components are.[br][br][b]Question 2: [/b]We learned about a similar idea, when finding the slope between two points. What should the signs of the components be, when the vector is pointing:[br]-up and to the right[br]-up and to the left[br]-down and to the left[br]-down and to the right[br][br][b]Question 3: [/b]We can now write vectors in [b]component form[/b]. For our classes, this will look like: [math]\binom{x}{y}[/math], where [math]x[/math] is the x-component, and [math]y[/math] is the y-component. Draw three of your own vectors, clearly with an axis, in your notebook, and write each vector in component form.[br][br][b]Question 4: [/b]Make the zero vector, and show the components. Hence, write the zero vector in component form.[br][br][b]Question 5: [/b]Describe how to find the magnitude of a vector, if you know its components. Hence, if [math]u=\binom{x}{y}[/math], find a formula for [math]\left|u\right|[/math].
Parallel Vectors
The applet below shows two vectors: [math]u[/math] and [math]w[/math], where [math]w=t\cdot u[/math], for some scalar [math]t[/math]. You can change vector [math]u[/math], by adjusting the sliders for magnitude and direction, and you can change vector [math]w[/math] by adjusting the slider for [math]t[/math]. [br][br]Play around with the applet for a bit, until you understand the relationships. Then, answer the questions below.
[b]Question 1:[/b] We will first make vector [b]w[/b] a unit vector. The question is, what scalar do we multiply to [b]u[/b], to get a unit vector (in the same direction)?[br][b]a)[/b] What is a unit vector?[br][br][b]b) [/b]Set the magnitude of [b]u[/b] to 2. Then, adjust t, to make [b]w[/b] a unit vector. What t value is this?[br][br][b]c)[/b] Repeat the above process, for various magnitudes of [b]u[/b].[br][br][b]d)[/b] Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] a unit vector in the same direction as [b]u[/b].[br][br][br][b]Question 2:[/b] We will now try to generalize this idea. Say we want a vector in the same direction as [b]u[/b], but length k?[br][b]a)[/b] Set the magnitude of [b]u[/b] to be 5. Then, adjust t to make [b]w [/b]have magnitude 10 (with the same direction as [b]u[/b]. What t value is this?[br][br][b]b)[/b] Repeat the above process, for various magnitudes of [b]u[/b]. Also, change the magnitude you want [b]w[/b] to be in (instead of 10).[br][br][b]c) [/b]Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] have length k, in the same direction as [b]u[/b].[br][br][br][b]Question 3: [/b]We can also extend this, to parallel vectors.[br][b]a)[/b] What does it mean for two vectors to be parallel?[br][br][b]b)[/b] Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] have length k, [b]parallel[/b] to [b]u[/b].
Vector Projections
To introduce the scalar product, we will begin with vector projections.[br][br]If vector [math]v[/math] is the "ground", with the sun directly above, and vector [math]u[/math] is a stick in the ground, a "shadow" will be created from vector [math]u[/math] onto vector [math]v[/math]. This "shadow" is called the [b]projection of [math]u[/math][/b] [b]onto [math]v[/math][/b].[br][br]The applet below shows two vectors, which are placed at the same initial point (since we can freely move vectors). You can move around the points, and then use the slider to create the projection of [math]u[/math] onto [math]v[/math]. You can also push "always show" to make the projection not reset.[br][br]Answer the questions, and move around the points to check various cases, until you are convinced of the property.
[b]Question 1:[/b] If the two vectors are acute, what is the relationship between the projection and the vector [math]v[/math]? What if the two vectors are obtuse?[br][br][b]Question 2:[/b] When is the projection equal to the zero vector?[br][br][b]Question 3: [/b]When is the projection equal to [math]u[/math]?[br][br][b]Question 4:[/b] Let the angle between the two vectors be [math]\theta[/math], and the two vectors be acute. Let the length of vector [math]u[/math] have notation [math]\left|u\right|[/math]. Find the length of the projection. [br][br]Now, we created a projection vector using the two vectors, with the same direction as vector [math]v[/math]. Next, we will define the [color=#ff0000]scalar product[/color] ([math]u\cdot v[/math]) as the product of lengths of vector [math]v[/math] and the length of the projection of [math]u[/math] onto [math]v[/math].[br][br][b]Question 5:[/b] Using this definition and your answer to question 4, write out an equation for [math]u\cdot v[/math].