Transcript Component Vectors

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
Opposite
ø
Adjacent
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 function
3) Use algebra to solve for the desired
variable & plug in
4) 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 components
4)Draw a new set of vectors
5)Use Pythagorean theorem to get the
magnitude of the resultant vector
6)Use arctangent to get the angle of the new
vector
Vector d1
X component
Y component
adj = hyp cos Θ
opp = hyp sin Θ
adj = 36 cos34º
opp = 36 sin34º
adj = + 29.8 m
opp = +20.1 m
Vector d2
X component
Y component
opp = hyp sin Ø
adj = hyp cos Θ
opp = 23 sin64º
adj = 23 cos64º
opp = - 20.7 m
adj = +10.1 m
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
dTotal =
(9.1)2
+ (dy)2
+ (30.2)2
+ 912.04
dTotal =
82.81
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