Physically Adding Vectors
physically by drawing and measuring them.
In the real world, we need much greater accuracy. (However, you learned the important concept
of visualizing vector addition).
Adding Vectors MathematicallyIn this section, we will add the same vectors mathematically
. To do this, we first must resolve each vector into its horizontal and
vertical components.
(Note: In this diagram, the angle is represented by the Greek
letter θ
or 'theta'. In mathematics and science, angles are usually labeled as such).
Y = 3 * Sin(45°)
In this case, there is
X = 6 * Cos(90°)
Y = 6 * Sin(90°)
Y = 5 * Sin(150°)
Summing up the vertical components (the Y values):
We determine the
direction of
the resultant vector and the formula is:
ArcTangent (of Resultant Vector) = ArcTangent (of Resultant Vector)= -4.8086237976514 At this point we have to be
careful in choosing the correct angle for the resultant vector.head of the angle or vector points to quadrant
IV.) Of course we know when we added the vectors physically, we found that the head of the
vector must lie in Quadrant II. Since the tangent function repeats every 180° adding this to
-78.25222928, we get 101.74777070718541 degrees which does lie in Quadrant II and therefore this
is the correct angle.Another way to determine the quadrant in which the resultant vector points is to look at the arctangent formula. This formula divides the 'Y' value by the 'X' value. So if 'Y' and 'X' are positive, it is Quadrant I. If 'Y' is positive and 'X' is negative, (as is the case for this example) the vector head points to Quadrant II. So, we are certain that the answer is
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