Vector Addition as Net Effect

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Suppose that a particle undergoes two displacements, the first one represented as \(\overrightarrow a \), which is 4 meters towards the east, and the second one represented as \(\overrightarrow b \), which is 4 meters towards the north:

Magnitudes of two displacements

To calculate the total distance travelled by the particle, we simply add the magnitudes of the two displacements: \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = 8\) meters.

However, what is the net effect of the two displacements? The net effect is that relative to its initial position, the particle is at a distance of \(4\sqrt 2 \) meters in the north-east direction:

Net effect of magnitudes of displacements

Thus, the net displacement of the particle is \(4\sqrt 2 \) meters, north-east. Observe that:

  • the net distance travelled is a scalar quantity, and equal to \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = 8\) meters.
  • the net displacement is a vector quantity. Since it is the result of vectors \(\overrightarrow a \) and \(\overrightarrow b \) combined, we can represent the net displacement as \(\overrightarrow a  + \overrightarrow b \). The value of this net displacement is \(4\sqrt 2 \) meters, north-east. Note that this + sign in \(\overrightarrow a  + \overrightarrow b \) does not represent normal addition, in the way we think about addition. It represents vector addition – it is calculating the net effect of two vectors.

To put it differently:

  • . \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\). is your normal, scalar, addition. In this case, \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\, = \,8\) meters.
  • \(\overrightarrow a  + \overrightarrow b \) is vector addition, and is equal to the net effect of vectors \(\overrightarrow a \) and \(\overrightarrow b \). In this case, \(\overrightarrow a  + \overrightarrow b \)  \( = \,4\sqrt 2 \) meters, north-east.

Exampl 1: Consider three vectors:

\(\overrightarrow a \): 2 units, east

\(\overrightarrow b \): 2 units, north

\(\overrightarrow c \): 2 units, west


(i) \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\)         (ii) \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow c } \right|\)      (iii) \(\left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right|\)

(iv) \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right|\)       (v) \(\overrightarrow a  + \overrightarrow b \)

(vi) \(\overrightarrow a  + \overrightarrow c \)       (vii) \(\overrightarrow b  + \overrightarrow c \)        (viii) \(\overrightarrow a  + \overrightarrow b  + \overrightarrow c \)

Solution: We note that \(\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = \left| {\overrightarrow c } \right| = 2\) units, and thus the answer for each of (i), (ii) and (iii) is 4 units, while the answer for (iv) is 6 units.

(v) Now, let us determine \(\overrightarrow a  + \overrightarrow b \). Observe the following figure:

Net effect of magnitudes of displacements - example 1

Clearly, \(\overrightarrow a + \overrightarrow b = 2\sqrt 2 \) units, north-east.

(vi) The following figure shows \(\overrightarrow a \) and \(\overrightarrow c \):

Parallel lines - no net effect

What will be the net effect of these two vectors? Intuitively, it should be obvious to you that they will cancel each other out, and thus the result will be a vector of magnitude 0 units. However, no particular direction will exist for this vector. We can write this as follows: \(\overrightarrow a  + \overrightarrow c  = 0\) units, unspecified direction, or \(\overrightarrow a  + \overrightarrow c  = \vec 0\)

(vii) The following figure shows \(\overrightarrow b \) and \(\overrightarrow c \), and the net effect of \(\overrightarrow b \) and \(\overrightarrow c \):

Net effect of magnitudes of displacements - example 2

Clearly, \(\overrightarrow b  + \overrightarrow c  = 2\sqrt 2 \) units, north-west.

(viii) The following figure shows \(\overrightarrow a \), \(\overrightarrow b \) and \(\overrightarrow c \):

Net effect of magnitudes of displacements - example 3

What is the net effect of all the three vectors taken together? Since \(\overrightarrow a \) and \(\overrightarrow c \) cancel the effect of each other out, we have:

\(\overrightarrow a  + \overrightarrow b  + \overrightarrow c  = \overrightarrow b  = \,2\) units, north

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