Vector Addition Tutorial

For the vector addition CALCULATOR, click here. Vector Addition Tutorial

Scalar quantities have magnitude only. (Examples are temperature and volume).
Vector quantities have magnitude and direction. (Examples are velocity and force).

Physically Adding Vectors

In the below left diagram, we see 3 vectors with their associated magnitudes and angles. In order to add these, we always must connect vectors 'head to tail' and the resultant vector (which represents the vector sum) is drawn from the tail of the first vector to the head of the last vector (see right side of the diagram below). In this example, using a ruler and protractor, we are able to get a resultant vector of about magnitude 11 and an angle of 102°.

In this section, we physically added the vectors 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 Mathematically

(Note: In the diagram below, the angle is represented by the Greek letter θ or 'theta'. In mathematics and science, angles are usually labelled as such).

In this section, we will add the same vectors mathematically . To do this, we first must resolve each vector into its horizontal and vertical components. We see from the above formulas that
X = Horizontal Component = Magnitude * Cos (θ)
X= 3 * Cos(45°)
X= 3 * 0.707106781186547... = 2.12132034355964
Y = Vertical Component = Magnitude * Sin (θ)
Y = 3 * 0.707106781186547... = 2.12132034355964

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 * -0.866025403784439 =
-4.33012701892219
Y = 5 * Sin(150°) = 5 * .5 = 2.5

Now we sum up the horizontal components (the X values):
X sum = 2.12132034355964 + 0 -4.33012701892219 = -2.20880667536255

Summing up the vertical components (the Y values):
Y sum = 2.12132034355964 + 6 + 2.5 = 10.62132034355964

We determine the magnitude of the resultant vector by the Pythagorean Theorem:
Magnitude ² = X² + Y²
Magnitude ² = -2.20880667536255² + 10.62132034355964²
Magnitude ² = 4.87882692912616 + 112.812445840514
Magnitude ² = 117.69127276964
Magnitude = 10.848560861683

To determine the direction of the resultant vector the formula is:

ArcTangent (Resultant Vector) = (Ysum / Xsum)

ArcTangent (of Resultant Vector) =
10.62132034355964
-2.20880667536255)

ArcTangent (of Resultant Vector)= -4.8086237976514

Angle = 101.747770718541 Degrees

We have calculated the resultatnt vector's magnitude and direction but there is just one more thing to determine.
Because the tangent function repeats every 180°, be careful in choosing the correct angle for the resultant vector.

For example, the arc tangent of -4.8086237976514 actually has 2 answers - the other being 281.747770718541 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 above diagram, 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 281.747 value since we are sure that 101.747770718541 Degrees is in the range of Quadrant II.

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