Basic Vector Operations
Both a magnitude and a direction must be specified for a vector quantity, in contrast to a
scalar quantity which can be quantified with just a number. Any number of vector
quantities of the same type (i.e., same units) can be combined by basic vector operations.
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Graphical Vector Addition
Adding two vectors A and B graphically can
be visualized like two successive walks, with
the vector sum being the vector distance from
the beginning to the end point. Representing
the vectors by arrows drawn to scale, the
beginning of vector B is placed at the end of
vector A. The vector sum R can be drawn as
the vector from the beginning to the end point.
The process can be done mathematically by
finding the components of A and B,
combining to form the components of R, and
then converting to polar form.
Example of Vector
Components
Finding the components of vectors
for vector addition involves
forming a right triangle from each
vector and using the standard
triangle trigonometry.
The vector sum can be found by
combining these components and
converting to polar form.
Polar Form Example
After finding the components for the vectors
A and B, and combining them to find the
components of the resultant vector R, the
result can be put in polar form by
Some caution should be exercised in
evaluating the angle with a calculator because
of ambiguities in the arctangent on
calculators.
Combining Vector Components
After finding the components for the vectors
A and B, these components may be just
simply added to find the components of the
resultant vector R.
The components fully specify the resultant of
the vector addition, but it is often desirable to
put the resultant in polar form.
Resolving a Vector Into Components
Vectors are resolved into components by use
of the triangle trig relationships. You may
change the length or angle of the polar form
of the vector, and the components will be
calculated below.
For vector A=
at angle
degrees,
the horizontal component is
=
and the vertical component is
=
The input to the boxes for units is arbitrary;
they serve to emphasize that the process of
vector addition is independent of the units of
the vector.
Magnitude and Direction from
Components
If the components of a vector are known, then its
magnitude and direction can be calculated with the
use of the Pythagorean relationship and triangle trig.
This is called the polar form of the vector.
If the horizontal component is
= Ax
and the vertical component is
= Ay,
then the magnitude is
= (Ax2 +Ay2) 1/2
and the angle is
θ = tan-1 (Ay/Ax) degrees.
The input to the boxes for units is arbitrary; they
serve to emphasize that the process of vector addition
is independent of the units of the vector.
Some caution should be exercised in evaluating the
angle with a calculator because of ambiguities in the
arctangent on calculators.
Vector Addition, Two Vectors
Vector addition involves finding vector
components, adding them and finding the
polar form of the resultant.
The addition of vector
A= 50 N at 30 degrees,
and vector
B= 100 N at 60 degrees,
yields components:
Rx = 50 Cos 30 + 100 Cos 60 = 93.3 N
Ry = 50 Sin 30 + 100 Sin 60 = 111.6 N
The resultant has magnitude
R = 145.463 N
and angle
θR = tan-1(Ry/Rx) = 50.1 degrees.
Also, the resultant can be written as: R = Rx i + Ry j = (93.3 i + 111.6 j ) N
Vector Addition, Three Vectors
Vector addition involves finding vector
components, adding them and finding the
polar form of the resultant.
The addition of vectors
A = 50 N at 0 degrees,
B = 40 N at 120 degrees, and
C = 30 N at 60 degrees
yields components:
Rx = 50 + 40 Cos 120 + 30 Cos 60 = 45 N
Ry = 0 + 40 Sin 120 + 30 Sin 60 = 60.62 N
The resultant has magnitude
R = (452 + 60.622)1/2 = 75.5 N
and angle
θR = 53.4 degrees.
Vector Addition, Four Vectors
(Do this example yourself.)
Vector addition involves finding vector
components, adding them and finding the
polar form of the resultant.
The addition of vectors
A=
at
degrees,
B=
at
degrees,
C=
at
degrees,
D=
at
degrees,
yields components:
+
+
+
=
+
+
+
=
The resultant has magnitude
R=
and angle
=
degrees.