In the applet below,[br][br]1) Note vectors [b]u[/b] and [b]v[/b] shown geometrically. (You can move them around by dragging the [b][color=#ff00ff]LARGE PINK POINTS[/color][/b] or by moving the vectors themselves). You can zoom in/out if you need to.[br][br]2) Move the [b]LARGE WHITE POINTS[/b] and/or drag the [b]THICK BLACK VECTOR[/b] around to create a vector that is the [b]resultant[/b] = sum of[b] u[/b] and [b]v[/b]. In order for this to happen you need the large pink for pink dot on vector V and match it with the small do for vector U. Then move the white dots onto the end points with the arrow side matching up with the arrow on vector V.[br][br]3) Select the [b]Check Resultant [/b]check box to check your answer. [br][br][br]
Thinking about your work above with the applet, can you come up with a generic expression about adding the vectors together?
When you add the vectors together you are really adding their x coordinates and then adding their y coordinates. So the equation is( Xu + Xv, Yu + Yv)
These are the component forms of vectors[math]\binom{\longrightarrow}{e}[/math] and [math]\binom{\longrightarrow}{f}[/math][br][math]\binom{\longrightarrow}{e}=[/math] (3,5)[br][math]\binom{\longrightarrow}{f}[/math]=(1, -6)[br]Add the vectors. (Your answer should be in ordered pair form.)[br][math]\binom{\longrightarrow}{e}[/math]+[math]\binom{\longrightarrow}{f}[/math] = [br]
Correct answer: (3 + 1, 5 + -6) = (4, -1)
These are the component forms of vectors[math]\binom{\longrightarrow}{c}[/math] and [math]\binom{\longrightarrow}{d}[/math][br][math]\binom{\longrightarrow}{c}=[/math] (5, -4)[br][math]\binom{\longrightarrow}{d}[/math]=(-2, 3)[br]Add the vectors. (Your answer should be in ordered pair form.)[br][math]\binom{\longrightarrow}{c}[/math]+[math]\binom{\longrightarrow}{d}[/math] = [br]
The correct answer is: (5 + -2, -4 + 3) = (3, -1)
These are the component forms of vectors[math]\binom{\longrightarrow}{a}[/math] and [math]\binom{\longrightarrow}{b}[/math][br][math]\binom{\longrightarrow}{a}=[/math] (3, -1)[br][math]\binom{\longrightarrow}{b}[/math]=(2, 3)[br][br]To get the answer to [math]\binom{\longrightarrow}{a}[/math]+[math]\binom{\longrightarrow}{b}[/math]...[br]Add their x values together and their y values together. Then just write then as an ordered pair as an answer.[br]
These are the component forms of vectors[math]\binom{\longrightarrow}{c}[/math] and [math]\binom{\longrightarrow}{d}[/math][br][math]\binom{\longrightarrow}{c}=[/math] (5, -4)[br][math]\binom{\longrightarrow}{d}[/math]=(-2, 3)[br]This time subtract the vectors. (Your answer should be in ordered pair form.)[br][math]\binom{\longrightarrow}{c}[/math]-[math]\binom{\longrightarrow}{d}[/math] = [br]
This time subtract the x values and y values.[br](5 --2, -4 - 3) = (7, -7)
What is one thing that you notice about this applet and how it compares to the other two applets?
Do you think multiplying two vectors, would work the same as adding or subtracting them? Explain.
Multiplication is a whole new game, we need to use matrix multiplication in order to find the solution as you saw with the applet above.
These are the component forms of vectors[math]\binom{\longrightarrow}{e}[/math] and [math]\binom{\longrightarrow}{f}[/math][br][math]\binom{\longrightarrow}{e}=[/math] (2, 3, 2)[br][math]\binom{\longrightarrow}{f}[/math]=(-5, 4, 3)[br] Multiply the vectors. [br][math]\binom{\longrightarrow}{e}[/math]*[math]\binom{\longrightarrow}{f}[/math] = [br]