1637 vector mathematics ap
TRANSCRIPT
Vector Mathematics
Adding, Subtracting, Multiplying and Dividing
Why?
• One can add 23 kg and 42 kg and get 65 kg.
• However, one cannot add together 23 m/s south and 42 m/s southeast and get 65 m/s south-southeast.
• Vectors addition takes into account adding both magnitude and direction
Words
• Vector: A measured quantity with both magnitude (the how big part) and direction
• Scalar: A measured quantity with magnitude only
• Resultant Vector: The final vector of a vector math problem
“Math” Coordinate System (Direction)
0º
90º
180º
270º
Polar Coordinate System (Direction and Magnitude)
Polar “Math” (Cartesian)
2 2
1
cossin
tan
r x yx ry r
yx
x
yr
θ
Vector addition
• Two Ways:1. Graphically: Draw vectors to scale, Tip
to Tail, and the resultant is the straight line from start to finish
2. Mathematically: Employ vector math analysis to solve for the resultant vector
Graphically 2-D Right
• A = 5.0 m @ 0°• B = 5.0 m @ 90°• Solve A + B
R
Start
R=7.1 m @ 45°
Important
• You can add vectors in any order and yield the same resultant.
Let’s add the last one mathematically
• The math you used previously doesn’t work (and I won’t let you use the Law of Sines or Cosines) or does it???
• What we will do is break each vector into components
• The components are the x and y values of the polar coordinate (go back 6 slides)
• Check out the next slides…
Components of Vectors
• A = Ax + Ay
• Ax =A cos θ
• Ay = A sin θ
• As long as you draw the x component first
A
Ax
Ay
θ
The Table Method
• We will organize these components in a table.
• See the board for this part and next slide
Table Method Equation
• Add all X components together Final Rx
• Add all Y components together Final Ry
Subtracting Vectors
• Simply add or subtract 180° (keep θ between 0° and 360°) to the direction of the vector being subtracted
• You just ADD the OPPOSITE vector (there is no subtraction in vector math)
Subtracting Vectors
Unit Vectors
• A unit vector is a vector that has a magnitude of 1, with no units.
• Its only purpose is to point• We will use i, j, k for our unit vectors• i means x – direction, j is y, and k is z• We also put little “hats” (^) on i, j, k to show
that they are unit vectors (I will boldface them)
Unit Vectors for vectors A & B
Unit Vectors
Adding using unit vectors
• R = A + B• R = (Ax + Bx )i + (Ay + By )j + (Az + Bz )k
which becomes R = Rx i + Ry j + Rz k
• The magnitude of R is found by applying the Pythagorean theorem
Multiplying Vectors (products)3 ways
1. Scalar x Vector = Vector w/ magnitude multiplied by the value of scalar
A = 5 m @ 30°3A = 15m @ 30°
Multiplying Vectors (products)
2. (vector) • (vector) = ScalarThis is called the Scalar Product or the
Dot Product
Dot Product Continued (see p. 25)
Φ
A
B
Multiplying Vectors (products)
3. (vector) x (vector) = vectorThis is called the vector product or the
cross product
Cross Product Continued
Cross Product Direction and reverse
Cross Product
• You can also solve the Cross Product with a matrix and unit vectors…check out the board for this.