Vectors FOR SAILSIM B Y E LLERY C HAN, PRECISION L IGHTWORKS 18 MARCH 2017

Transcription

1 Vectors FOR SAILSIM B Y E LLERY C HAN, PRECISION L IGHTWORKS 18 MARCH 2017

2 Despicable Me! helped me with my math homework -- comment on an online forum

3 Vectors Useful for representing values that have a direction and an amount Force Velocity Hide that you are really doing messy coordinate calculations 3 m/s 0.01 m/s

4 What is a Vector? An arrow That has a specific fixed length That points in a specific fixed direction That has no specific location These vectors are all equivalent

5 Adding Vectors To determine the net (total) force acting on something Add up individual forces Tug-of-War Team A 2 turtle power 3 turtle power Team B + 2 tp 3 tp Start here = 1 tp End here It s 2 against 3. The net force is 1 tp pulling to the right, so Team B wins.

6 More Vectors You can compute the net result of multiple forces the same way. Add the force vectors together to get a net force Team A 3 turtle power Team B Tug-of-War Each side gets another player. Who will win now?

7 Adding Multiple Vectors This works with multiple vectors. Add them by Putting the vectors end-to-end Drawing the vector from the start to the finish 3 tp Team A Team B Team A wins, even though it was 3 against 4!

8 Scaling a Vector Add 3 copies of a vector to get a vector 3 times as long Changes the magnitude but not the direction V + V + V = 3 V What if you scaled V by -1?

9 Components of a Vector

10 Magnitude of a Vector Sometimes we want just the length of a vector How far is it, as the crow flies? This is called the vector s magnitude The magnitude of V is written V 4 miles 5 miles 3 miles

11 Direction of a Vector Sometimes we want just the direction of a vector Can we remove the magnitude of a vector and leave the direction? Remember that the magnitude of V is V Drill at this angle to reach the destination tunnel entrance tunnel exit

12 Unit Vector Sometimes it is useful to take the magnitude out of a vector That would leave just the direction We can t really do that: it wouldn t be a vector A vector with a length of zero is not useful It would be a dot, and we could not tell what direction it is pointing > However, a vector of length one (a unit vector) is very useful Divide the vector by its length (scale it by! ) to get a unit vector # THIS WAY TO EXIT?? # # = U direction(u) = direction(v); U = 1 divide V into 3 parts

13 Maps Vectors can be very useful on maps Start at Go East for 2 miles Go Northwest for 1 mile Find treasure! Remember, we said vectors have no location, so how did we find the treasure? Ye are here

14 Points and Vectors Points have location And nothing else We start at and the vectors provide offsets from that point There are some simple rules: Vector + Vector = Vector Point + Vector = Point Point + Point not allowed Point Point = Vector

15 Computing with Vectors How do we get vectors into the computer so they can be useful? Ow! We need a different representation

16 Vector Components C can be decomposed into A and B A and B are component vectors that define C A is the portion of C in the X direction B is the portion of C in the Y direction A and B are called orthogonal components because they are perpendicular to each other A + B = C A C B B A

17 Vector Components Before, we separated a vector into its magnitude and direction components We can also break it down differently One way is to place it on a Cartesian grid Now it is easy to see how to break it into a part parallel to the X axis, and a part parallel to the Y axis We can describe C as the ordered pair [4, 7] It is the sum of A [4, 0] and B [0, 7] B y C A x

18 Maps Again Maps often use Cartesian coordinates Now we can get the coordinates of the treasure Instruction East North Ye are here Go East 1000 steps = Go Northwest 400 steps Ye are here 283 steps 283 steps Go North 283 steps = 783 Go West 283 steps = C 1000 steps Findye treasure How can you figure out that 400 steps NW = 283 steps N steps W?? B Use the Pirate-thagorean Theorem A

19 Adding Vectors In general: Add vectors by adding their corresponding components V = A + B v x, v y = a x, a y + b x, b y = a x +b x, a y +b y

20 Vector Inverse The inverse of a vector is a vector of the same length (magnitude) pointing in the opposite direction Remember scaling a vector by a negative number? W = -V = -1 x V V V = (5, 3) W = -1 x (5, 3) = (-1 x 5, -1 x 3) = (-5, -3) W

21 Vector Subtraction With the vector inverse, we can subtract vectors B and C are component vectors that define A B is the portion of A in the X direction C is the portion of A in the Y direction y -A C = A + B C A = C + ( A) = B C B = C + ( B) = A -B B C A x

22 Vector Magnitude The length or magnitude of a vector is the distance from its start to its end Remember the distance formula? v s = 0, 0 v e = 5, 3 V = d = x 2 + y 2 = (5 0) = = = v s V v e length = 5.83

23 Unit Vectors A unit vector is any vector with a length of 1 Make a unit vector by dividing any vector by its length y U = V V = 1 V V V = 4, 7 V = 4 H + 7 H = = 65 V U = 1 65 V = , = [0.496, 0.868] x

24 Car Meets Wall When the car hits the wall Will the wall break? If not How far will the car scrape along the wall? assuming the wall doesn t break we will have to know something about friction How do we get from A (car velocity) and B (wall direction) to S (velocity to wall) and W (velocity to wall)? A B B is a unit vector since we only know its direction S is the portion of A. in the direction. of B. B W W is the portion of A. perpendicular to the direction of B S A

25 Vector Projection Sometimes we want to know how much influence a vector applies in a given direction In this case, what portion of A goes in the direction of B? The direction may not be aligned with the X or Y axis We know A and B. How do we find S and W? Trigonometry (but we would need to know some angles) B Vector projection (S is the image of A projected onto B) Vector dot product No angles, sines, or cosines needed! S is the portion of A. in the direction. of B. W W is the portion of A. perpendicular to the direction of B S A

26 Dot Product The vector dot product takes two vectors and returns a scalar (a number) Projects one vector onto another Finds the magnitude of the component of a vector in a given direction No trigonometry, just simple arithmetic! d = A Q B = a x, a y Q b x, b y = a x b x + a y b y

27 Back to the Car Problem 1. Project A onto B using the Dot Product to get the length of S 2. Scale the unit vector B to get the vector S 3. Subtract S from A to get W B S = A Q B S = S B S is the portion of A. in the direction. of B. W W is the portion of A. perpendicular to the direction of B S 3. W = A S A

28 Finishing the Problem Now we have enough information about the car s motion to help solve the problem Will the wall break? We have the velocity vector W that we can use with the car s mass to figure out how much force will be applied against the wall How far will the car scrape along the wall? We have the velocity vector S that we can use along with a coefficient of friction to figure out how far the car will slide along the wall B W A S

29 Vectors and Beyond This discussion only talked about 2D vectors 3D vectors get used a lot, too, in physics, electronics, and computer graphics Higher-dimensional vectors get used in AI and big data analysis Other things to Google: Vector rotation Vector cross product If you want to learn computer graphics or game programming Homogeneous coordinates (using 4D vectors) Transformation matrices

30 License Copyright 2017, Ellery Chan and The SailSim Team SailSim 2017 is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License