Physics 30
Finding Resultant Vectors Using Law of Cosines and Law of Sines
We know from previous lessons that we can find the resultant vector of
a right triangle using Pythagorean’s theorem. The problem is that not
all motion will be moving in perpendicular motion or from right
triangles. We need to have solutions for solving for resultant vectors
that are not right triangles.
We will use the law of cosines and the law of sines.
Law of Cosines:
The Law of Cosines is a general equation relating three sides and one
angle in a triangle. There are no restrictions on the triangle's shape.
Three elements determine a triangle. If any three of the four elements
in the law-of-cosines equation are given, the equation allows you to
calculate the fourth one.
There are three law-of-cosines equations, depending on which angle is
included:
c2 = a2 + b2 - 2ab cos g
(1)
a2 = b2 + c2 - 2bc cos a
(2)
b2 = c2 + a2 - 2ca cos b
(3)
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Law of Sines:
The Law of Sines is a set of equations true for any triangle. It states that
the ratio "sine of an angle divided by the length of the opposite side" is
the same for any pair of angle and opposite side.
The law-of-sines equations are
A triangle is determined by three of its elements. Given two sides and
an angle opposite to one of the sides, the Law of Sines lets you
determine the angle opposite to the other side.
Let’s try an example:
Vector 1 is 100m, while vector 2 is 130m. Find vector r.
The Law of Cosines can be used to calculate the magnitude r of the
resultant vector.
Physics 30
The Law of Sines can then be used to calculate the direction (q) of the
resultant vector.
Components of Vectors
Vectors can be described in terms of their scalar components. A vector
in two dimensions has two scalar components, one along the x-axis and
one along the y-axis. For a vector , these
components are denoted ax and ay,
respectively.
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The vector in Figure 7 has a magnitude of 8 and an angle q with the
positive x-axis equal to 30o. Its scalar components have the values
ax = 6.93,
ay = 4.00
The definition of the sine and cosine imply that
ax = a cos θ,
ay = a sin θ
Substituting a = 8.00 and q = 30.0o into these equations gives the
answers 6.93 and 4.00
Let’s try an example!
The magnitude and direction of
are:
v1 = 100, θ1 = 60o
v2 = 130, θ2 = 140o
1
and
2
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The components of each vector are calculated using the appropriate
trigonometric functions.
Vector
Diagram
x - component
y - component
To add the two vectors is to add the respective components. If the
components of the resultant are denoted (rx,ry), we get:
Physics 30
rx = v1x + v2x
rx = (+50.00) + (-99.59)
rx = -49.59
ry = v1y + v2y
ry = (+86.60) + (+83.56)
ry = +170.16
You could specify the resultant in terms of the components and stop
the calculation at this point. However, if the magnitude and direction
angle of the resultant are required, these can be calculated from the
components as follows:
The direction can be calculated using the definition of the tangent of an
angle
This implies for the direction angle of the value 180 73.75 = 106.24o
Physics 30
In summary, if two vectors 1 and 2 are given in terms of magnitude
and direction, a resultant can be calculated by doing the following:





Use a vector diagram and trigonometric functions to convert the
vectors to component form.
Add the components (xtotal = x1 + x2) and (ytotal = y1 + y2).
Remember to include positive or negative directions.
Draw the resultant vector using the xtotal component and the ytotal
component. Remember to include positive or negative
directions.
Calculate the resultant magnitude using the Pythagorean theorem
(c2 = a2 + b2).
Calculate the direction using the appropriate trigonometric
function (tangent function).
Practice
1. Using the component method, calculate the resultant (sum) of the following two vectors. Show
all required calculations and diagrams below and identify the direction
= 175, 70o
o
2 = 200, 200
1
a) Vector diagram and calculation of the
components for
1.
b) Vector diagram and calculation of the
components for
2.
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c) Addition of the components and drawing of the resultant vector.
d) Calculation of the resultant magnitude using the Pythagorean theorem.
e) Calculation of the resultant direction using the tangent function. Express the direction.
2.
= 185, 45o
o
2 = 95, 320
1
a) Vector diagram and calculation of the
components for
1.
b) Vector diagram and calculation of the
components for
2.
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c) Addition of the components and drawing of the resultant vector.
d) Calculation of the resultant magnitude using the Pythagorean theorem.
e) Calculation of the resultant direction using the tangent function. Express the direction.