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

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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