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 = +1

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4

How to show magnitude of vectors - mathematically and graphically

Adding two vectors graphically

A + B = R

Head to tail methodHead to tail method

Showing A + B = B + A

Showing A - B ≠ B - A

Tail to tail methodTail to tail method

Showing A - B = A + (- B)

Breaking vectors down in component parts

V = Vx + Vy + Vz

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 + ByBx = Cos 60º (20) = 10By = Sin 60º (20) = 17.3

Adding Vectors mathematically

Step 2: Solve for Vx an Vy

Vx = Rx = Ax + Bx = 8.67 + 10 = 18.67

Vy = Ry = Ay + By = 5 + 17.3 = 22.3

Step 3: Solve for R (magnitude)

|R|2 = Vx2 + Vy2

|R|2 = 18.672 + 22.32

|R|2 = 348.57 + 497.29 = 845.86

|R| = (845.86)1/2

|R| = 29.1

Step 4: Solve for an angle

Tan (Vector Angle - from x axis) = 22.3/18.67 = 1.194

Tan -1 (1.194) = 50.1º

A

B

Graphical CheckGraphical Check

B

A

10

10

10

A

B

A

A + BB

AAy = 5

Ax = 8.67

By = 17.3

Bx = 10

B

Ry = 17.3 + 5 = 22.3

Rx = 8.67 + 10 = 18.67

R = A + B = 29.1

Angle = Tan -1 22.3/18.67 = 50.1º