Vectors Angle Reference direction

Vector A is identical to Vector B, just transported (moved on a graph keeping the same orientation and length). Vector A Vector B

Cartesian CCW = + Compass CW =

How to show magnitude of vectors - mathematically and graphically

Adding two vectors graphically A + B = R Head to tail method

Showing A + B = B + A

Showing A - B ≠ B - A Tail to tail method

Showing A - B = A + (- B)

Breaking vectors down in component parts

V = V x + V y + V z

Step 1: Break down vectors to be added into there Vx and Vy components (for three dimension x, y and z components) Step 2: Sum the Vx and then Vy components. Step 3 use the Pythagorean theorem to solve for the magnitude resultant vector Step 4: Use SOH-COA-TOA to find the vector angel from the x axis Example: Add vector A =10 that points to 030º (Cart) with a vector B = 20 that points to 060º (Cart) A B Step 1: Break vectors into components A = Ax + Ay Ax = Cos 30º (10) = 8.67 Ay = Sin 30º (10) = 5 B = Bx + By Bx = Cos 60º (20) = 10 By = Sin 60º (20) = 17.3 Adding Vectors mathematically

Step 2: Solve for Vx an Vy Vx = Rx = Ax + Bx = = Vy = Ry = Ay + By = = 22.3 Step 3: Solve for R (magnitude) |R| 2 = Vx 2 + Vy 2 |R| 2 = |R| 2 = = |R| = (845.86) 1/2 |R| = 29.1 Step 4: Solve for an angle Tan (Vector Angle - from x axis) = 22.3/18.67 = Tan -1 (1.194) = 50.1º

A B Graphical Check B A 10

A B A A + B B A Ay = 5 Ax = 8.67 By = 17.3 Bx = 10 B Ry = = 22.3 Rx = = R = A + B = 29.1 Angle = Tan /18.67 = 50.1º