Adding Vectors Graphically CCHS Physics
Dec 31, 2015
Adding Vectors Graphically
CCHS Physics
Vectors and Scalars
• Scalar has only magnitude
• Vector has both magnitude and direction– Arrows are used to represent vectors– The direction of the arrow gives the
direction of the vector– The length of a vector arrow is proportional
to the magnitude of the vector
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Vector Properties
• Notation– When vector is handwritten, often shown
with arrow or other designation – In book, usually bold face type, ex: A– Magnitude of A represented by italic, ex: A
• Equality of Vectors– Two vectors, A and B, are defined as equal
if they have the same magnitude and direction
A→
or A or A
Vector Properties Cont.• Vector Addition (graphically)
– All the vectors must have the same units– Tip-to-Tail Method of Addition
• Draw vector A to scale (ie 1 cm = 1 m)• Then draw vector B to the same scale with the
tail of B starting at the tip of A• Resultant vector R is given by R = A + B
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Vector Properties Movie
Vector Properties Cont.
– Parallelogram Method of Addition• The tails of vectors A and B
are joined, and the resultant vector, R, is the diagonal of the parallelogram formed with A and B as its sides.
– Note A + B = B + A– To add more than two
vectors, just continue adding tail to tip.
Vector Properties
– Negative of a Vector• When a vector is multiplied by -1, the
magnitude of the vector remains the same, but the direction is reversed
Vector Properties Cont.
• Vector Subtraction (graphically)– Carried out exactly like vector addition,
except that one of the vectors is multiplied by a scalar factor of -1
– A - B = A + (- B)
Subtracting Vectors
Vector Properties Cont.
• Multiplication and Division of Scalar by Vectors– Multiplication or division of vector by a
scalar yields a vector– If the given vector B is multiplied by the
scalar 4, the result, written 4B, is a vector with a magnitude four times the original vector B, pointing in the same direction as B.
B 4B
Multiplication by Scalar
Adding Vectors Graphically
Displacement Hike4 km East
2 km NE
3 km @ 120°
5 km @ 210°
R = 1.6 km @ 105°
START
FINISH
Scale: 1” = 1km
ADDING VECTORS MATHEMATICALLY
CCHS PHYSICS
Components
• Components: projections of a vector along axes of rectangular coordinate system– Can resolve vectors into components
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Trigonometry Review
• h = length of hypotenuse of right triangle• ho = length of side opposite the angle • ha = length of side adjacent to the angle
sin =ho
h
cos =ha
h
tan =ho
ha
Trig Review Cont.
• Inverse Trigonometric Functions
• Pythagorean Theorem
=sin−1 hoh
⎛⎝⎜
⎞⎠⎟
θ = cos−1 hah
⎛⎝⎜
⎞⎠⎟
θ = tan−1 hoha
⎛
⎝⎜⎞
⎠⎟
A = Ax2 + AY
2
Finding Vector Components
Adding Vectors Mathematically
• Select coordinate system• Resolve all the vectors into components• Add all the x-components• Add all the y-components
– The sum of the x and y components gives you the components of the resultant
• Find the magnitude of the resultant via the Pythagorean Theorem
• Find the angle with a suitable trig function
Adding Mathematically
Displacement Hike Revisited
Given the following vectors, mathematically determine the resultant:
4 km East
2 km NE
3 km @ 120°
5 km @ 210°
Displacement Hike Revisited
Action X Component Y Component
4 km E x = 4 km y = 0 km
2 km NE x = 2cos45x = 1.4 km
y = 2sin45y = 1.4 km
3 km @ 120°
x = -3cos60= -1.5 km
y = 3sin60= 2.6 km
45°
2 km
4 km
120°3 km
60°
Displacement Hike Revisited
Action X Component Y Component
5 km @ 210°
x = -5cos30x = -4.3 km
y = -5sin30y = -2.5 km
RESULTANT x = 4+1.4-1.5-4.3=-0.4 km
y = 0+1.4+2.6-2.5=1.5 km
Magnitude:
Direction:
−0.4( )2
+ 1.5( )2
= 1.6km
tan =1.5.4
→ =75°→ =105°
210°
5 km
30°
105°
R = 1.6 km
75°
Rx = -.4
Ry
= 1
.5
Resultant and Equilibrant• Resultant: the single vector (usually with regards
to force) that is equal to two or more other vectors• Equilibrant: the single vector (usually with regards
to force) that will balance two or more vectors– Equal in magnitude opposite in direction to the resultant
F1
F2 Resultant
Equilibrant
45°
2 km
210°
5 km
30°
F1
F2 Resultant
Equilibrant
120°3 km
60° 105°
R = 1.6 km
75°
Rx = -.4
Ry
= 1
.5