Scalar

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Scalar
a quantity described by magnitude only
examples include
time, length, speed, temperature, mass, energy
Vector
a quantity described by magnitude and direction
examples include
velocity, displacement, force, momentum,
electric and magnetic fields
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Vectors are usually named with capital letters
with arrows above the letter.
They are represented graphically as arrows.
The length of the arrow corresponds to the
magnitude of the vector.
The direction the arrow points is the vector
direction.
Examples include
A 20 m/s at 35 NE
B 120 lb at 60 SE
C 5.8 mph/s west
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Vector Addition
vectors may be added graphically or analytically
Triangle (Head-to-Tail) Method
1. Draw the first vector with the proper length
and orientation.
2. Draw the second vector with the proper length
and orientation originating from the head of
the first vector.
3. The resultant vector is the vector
originating at the tail of the first vector
and terminating at the head of the second
vector.
4. Measure the length and orientation angle of
the resultant.
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Example
A 11 N _at_ 35 NE
Find the resultant of A and B.
B 18 N _at_ 20 NW
B
20 NW
R
R 14.8 N _at_ 57 NW
A
35 NE
57 NW
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Parallelogram (Tail-to-Tail) Method
1. Draw both vectors with proper length and
orientation originating from the same point.
2. Complete a parallelogram using the two
vectors as two of the sides.
3. Draw the resultant vector as the diagonal
originating from the tails.
4. Measure the length and angle of the
resultant vector.
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Explore more vectors at link, link, link, and
link.
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Resolving a Vector Into Components
y
The horizontal, or x-component, of A is found
by Ax A cos q.
A
Ay
q
Ax
The vertical, or y-component, of A is found by
Ay A sin q.
x
By the Pythagorean Theorem, Ax2 Ay2 A2.
Every vector can be resolved using these
formulas, such that A is the magnitude of A,
and q is the angle the vector makes with the
x-axis.
Each component must have the proper
sign according to the quadrant the vector
terminates in.
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Analytical Method of Vector Addition
1. Find the x- and y-components of each vector.
Ax A cos q
Ay A sin q
By B sin q
Bx B cos q
Cx C cos q
Cy C sin q
Rx
Ry
2. Sum the x-components. This is the
x-component of the resultant.
3. Sum the y-components. This is the
y-component of the resultant.
4. Use the Pythagorean Theorem to find the
magnitude of the resultant vector.
Rx2 Ry2 R2
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5. Find the reference angle by taking the inverse
tangent of the absolute value of the
y-component divided by the x-component.
q Tan-1 Ry/Rx
6. Use the signs of Rx and Ry to determine the
quadrant.
NE
NW
(,)
(-,)
(-,-)
(-,)
SW
SE