Chapter 3 – Vector Addition Notes/Examples
What is a Vector?
Vectors are graphic representations of physical occurrences. We use vectors to
visualize what physically happens to an object under certain conditions. Vectors
designate the magnitude and direction of things like forces, accelerations, velocities,
and displacements. They help us to describe motion through a common symbology.
Magnitude
-
the amount of the force, acceleration, velocity, or displacement.
Direction
-
a specific direction in which the force is applied, or the
acceleration, velocity, or displacement occur.
What is Vector Addition?
Vector addition is the process of finding the resultant vector when given the
components of the vector.
In one-dimensional vector addition, you will be working on either the x
(horizontal) or y (vertical) axis. The vector addition is simply the addition (or subtraction
if the vectors are in opposite directions) of the two vectors. If they are in opposite
directions, the resultant direction will be in the same direction as the vector with the
greatest magnitude.
In two-dimensional vector addition, you have both an x (horizontal) and a y
(vertical) component. You add the vectors by using the Pythagorean theorem since the
two vectors are at right angles to each other. This value is the magnitude of the result
vector. The direction of the result vector is the angle between the hypotenuse of the
triangle and the horizontal. To find this value, you use the tan-1 function on the
calculator.
What is Vector Resolution?
Vector resolution is used in two-dimensional vector analysis and is the reverse to
vector addition. In vector resolution, you know the magnitude and direction (angle) of
the resultant vector and you are solving for the horizontal and vertical components.
To find the x (horizontal) component, you use the cosine function and the formula
if the angle is with respect to the x-axis:
Rx  R cos ; where R is the magnitude of the resultant
To find the y (vertical) component, you use the sine function and the formula:
Ry  R sin  ; where R is the magnitude of the resultant
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Chapter 3 – Vector Addition Notes/Examples
Addition of Vectors in Two Dimensions Worksheet
Right Angles use the following
formulas:
Obtuse and Acute angles use the
following formulas:
a2 + b2 = c2
a2 + a2 – 2a·b·cos(C) = c2
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
sin(A) = sin(B) = sin(C)
a
b
c
Examples:
1. A plane flies 122m/s south through a westerly wind going 35.0m/s. What is the
resultant velocity of the plane?
2. A boat going 23.5m/s east crosses a river flowing 12.0m/s south. What is the
resultant velocity? If the river in number 3 was 235m wide, how far downstream
would the boat travel before hitting the other side?
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Chapter 3 – Vector Addition Notes/Examples
3.
12N
28N
35
East
Find the vector sum of the forces above. Include both the magnitude and a
description of the direction in your answer.
4. A car drives 3.50 x 102 m 53° SoE and then turns and travels 7.50x102 m S 64°
WoN. What is the displacement of the car? What is the distance the car traveled?
5. Mr. Schmidt needs to move a very large china cabinet into his house. He employs
the use of three of his friends. Each friend pulls on the cabinet in the following
fashion.
F1 = 16.2 N @ 48.0o (or 48.0o EoN)
F2 = 39.6 N @ 297.0o (or 27.0o NoW)
F3 = 11.2 N @ 356.0o (or 4.00o WoN)
What would be the direction and magnitude of the resulting force acting on the
cabinet?
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