How to Add Vectors

On this page, His Nibs Herkes, will add the some vectors mathematically . To do this, you first must draw each vector into its horizontal and vertical components.  Remember to set your calculators to round through the hundreds place!  In the following picture, the angle is shown with the Greek letter  'Θ - theta - an angle':
/tmp/lyx_tmpdir9441f6SwEJ/lyx_tmpbuf0/0_home_dherkes_downloads_vector1.gif
       X = Horizontal Component = Magnitude * Cos ( Θ )
        X= 3 * Cos(45°)
        X= 3 * 0.707 = 2.121
        Y = Vertical Component = Magnitude * Sin ( Θ )
        Y = 3 * 0.707 = 2.12
    Moving on to the next vector:
        X = 6 * Cos(90°) = 0
        Y = 6 * Sin(90°) = 6
    And for the final vector:

        X = 5 * Cos(150°) = 5 * -0.866 = -4.330
        Y = 5 * Sin(150°) = 5 * .5 = 2.5

Now we will sum (Σ - sigma - to sum) the horizontal components (the X values):
  Σ X = 2.12 + 0 -4.330 = -2.209
Summing (Σ ) the vertical components (the Y values):
  Σ Y = 2.121 + 6 + 2.5 = 10.621
We determine the magnitude of the resultant vector by the Pythagorean Theorem:
Magnitude 2 = X2 + Y2  (Remember that the magnitude is just a scalar, thus is part of this vector.)
Magnitude 2 = -2.209 + 10.621
Magnitude 2 = 4.879 + 112.813
Magnitude 2 = 117.691
Magnitude = 10.849
To determine the direction of the resultant vector the formula is:
ArcTangent (Resultant Vector) = ( Σ Y / Σ Y)
ArcTangent (of Resultant Vector) = (10.621 / -2.208)
ArcTangent (of Resultant Vector)= -4.809
Angle = 101.748 Degrees /tmp/lyx_tmpdir9441f6SwEJ/lyx_tmpbuf0/1_home_dherkes_downloads_xy-graph.gif

We  just have calculated the  vector's magnitude and direction but there is  one more thing to be determined.
Because the tangent function repeats every 180°, be careful in choosing the correct angle for the vector. For example, the arc tangent of -4.809 actually has 2 answers - the other being 281.748 degrees. How do we know which to choose? In the above arc tangent calculation, we see that the 'Y' value is positive and the 'X' value is negative. Referring to the diagram below, when 'Y' is positive and 'X' is negative, the angle is in "Quadrant II" and the angle must fall within the 90° to 180° range. Therefore we can rule out the 281.748 value since we are sure that 101.748 Degrees is in the range of Quadrant II.

File translated from TEX by TTH, version 3.70.
On 17 Dec 2005, 11:26.