Component Vectors•Vectors have two parts (components)–X component – along the x axis

–Y component – along the y axis

Finding components

•X component – follow from the tail of the vector along the x axis until you reach the point where the tip would be if it fell straight down

•Y component – follow from where you stopped on the x axis straight up to the tip

•You should now have formed a right triangle with the original vector as the hypotenuse

                                                                                            

                                                                                            

                                                                                            

                                                                                            

To find components• To find components, you must use

trigonometric functions

Hypotenuse

Adjacent

Opposite

ø

Trig functions• Θ is the angle between

the vector and the x axis• sin Θ = _opposite_

hypotenuse• cos Θ = _adjacent_

hypotenuse• tan Θ = _opposite_

adjacent

Steps for finding the components

1) Draw a picture (arrowheads, original vector & components)

2) Choose a trig function3) Use algebra to solve for the

desired variable & plug in4) Calculator in degrees!5) Check with Pythagorean theorem

Example

•

                                                      

X component

•cos Θ = _adjacent_ hypotenuse

•cos 35 = _adjacent_ 316

•316 cos 35 = adjacent•259 N = adjacent

Y component

• sin Θ = _opposite_ hypotenuse

• sin 35 = _opposite_ 316

• 316 sin 35 = opposite• 181 N = opposite

How to find components when you add two vectors

1)Find the x and y component for both vectors

2)Add up the x components 3)Add up the y components4)Draw a new set of vectors5)Use Pythagorean theorem to get the

magnitude of the resultant vector6)Use arctangent to get the angle of the new

vector

                                                                             

                                             

Vector d1

X component

adj = hyp cos Θ

adj = 36 cos34º

adj = + 29.8 m

Y component

opp = hyp sin Θ

opp = 36 sin34º

opp = +20.1 m

X component

opp = hyp sin Ø

opp = 23 sin64º

opp = - 20.7 m

Y component

adj = hyp cos Θ

adj = 23 cos64º

adj = +10.1 m

Vector d2

Total X displacement – add d1 and d2

dtotal = d1 + d2

dtotal = 29.8 m + (-20.7m)

dtotal = +9.1m

Total Y displacement – add d1 and d2

dtotal = d1 + d2

dtotal = 20.1 m + 10.1m

dtotal = +30.2m

To get the magnitude of the resultant vector

• Use Pythagorean Theorem

dTotal = (dX)2 + (dy)2

dTotal = (9.1)2 + (30.2)2

dTotal = 82.81 + 912.04

dTotal = 994.85 = 31.5 m

To find the angle of the resultant vector

•Use arctangent function: Θ = tan-1 (opp/adj) Θ = tan-1 (30.2/9.1) Θ = tan-1 (3.3) Θ = 73.1°

Formulas• a2 + b2 = c2

• R2 = a2 + b2 - 2ab(cosθ)

• SOH• CAH• TOA