Download Resolution of Vectors into Rectangular Components

Resolution of Vectors into Rectangular Components
To break a vector, ~v , into rectangular components means to break them into
x and y (or horizontal and vertical components).
The horizontal and vertical components of ~v are normally expressed as
v~x and v~y respectively. The vector ~v
is always the sum of its components,
~v = v~x + v~y .
In order to solve for the magnitudes of v~x and v~y we must use trigonometry
for right angled triangles.
Example 1.1. Find v~x and v~y for the vector in the following diagram.
Since we have a right angle triangle,
cos 65◦ =
vx
28N
sin 65◦ =
vy
28N
⇒ vx = cos 65◦ · 28N
vy = sin 65◦ · 28N
⇒ v~x = 11.8 N [Right]
⇒ v~y = 25.4 N [Down].
By breaking vectors into their rectangular components, we can add vectors
in a new way. This method is very adaptable and useful in cases where the
sine and cosine law methods are cumbersome. It will also be used more
exclusively later in the course.
*Explore the GSP file
VectorAdditionByComponents.gsp
Assign the students questions from the text for them to practice breaking
vectors into components.
Example 1.2. Inclined Plane Example A box weighing 140 N is resting on
a ramp that is inclined at an angle of 20◦ . Resolve the weight vector F~g into
rectangular components that are parallel and perpendicular to the plane.
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In this case, we must break F~g into
rectangular components as shown in
the diagram on the right. However, in
order to calculate the magnitudes of
the components, we must first determine the angle θ.
From the next diagram we can see that
x + z = 90◦ since the sum of angles
in a triangle must be 180◦ , but it is
also clear that z + y = 90◦ . Thus,
x = y, and so the angle θ is in fact
the same measure as the angle of the
incline, and θ = 20◦ .
Now to calculate the components, we have that
g// = (140N ) sin 20◦
g⊥ = (140N ) cos 20◦
g// = 48 N
g⊥ = 132 N
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