Lesson video

Simulations

https://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles/Vector-Addition/Vector-Addition-Interactive

https://www.geogebra.org/m/rj9YPySM

Lesson objective

Dear Learners,

By the end of this section, you should be able to:-

• Describe the graphical method of vector addition;
• Describe the graphical method of vector addition;

Graphically, vectors can be added using the triangle, parallelogram and polygon method of vector addition. You can add or subtract vectors using the algebraic or graphical method of vector addition.

Key concepts

Graphically, vectors can be added using the triangle, parallelogram and polygon method of vector addition.

To determine the resultant of two vectors acting:
in the same direction, add the given vectors and take the common direction.
in opposite directions, get the difference and take the direction of the vector with the greater value

Brainstorming Question

If two vectors have equal magnitude, what are the maximum and minimum magnitudes of their sum?

Triangle method of vector addition

Triangle law of vector addition is used to find the sum of two vectors. This law is used to add two vectors when the first vector’s head is joined to the tail of the second vector and then joining the tail of the first vector to the head of the second vector to form a triangle, and hence obtain the resultant sum vector. That’s why the triangle law of vector addition is also called the head-to-tail method for the addition of vectors.

Thus, if two vectors acting simultaneously on a body are represented both in magnitude and direction by two sides of a triangle taken in an order, then the resultant vector (both magnitude and direction) of these two vectors is given by the third side of that triangle taken in the opposite order. This is the statement for the triangle law of vector addition.

Figure 1.3 The triangle rule for the addition of two vectors

Parallelogram method of vector addition

The vector addition may also be understood by the law of parallelogram. The parallelogram law of vector addition is used to add two vectors when the vectors that are to be added form the two adjacent sides of a parallelogram by joining the tails of the two vectors. Then, the sum of the two vectors is given by the diagonal of the parallelogram.

Thus, if two vectors are represented by the two adjacent sides (both in magnitude and direction) of a parallelogram drawn from a point, then their resultant vector is represented completely by the diagonal of the parallelogram drawn from the same point. This is the statement for the parallelogram law of vector addition.

Figure 1.4. The Parallelogram rule for the addition of two vectors.

Suppose two vectors $\overrightarrow{A} ~and~ \overrightarrow{B}$ are at arbitrary positions as shown in Figure 2 (a). Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of vector$\overrightarrow{A}$ you draw a line parallel to vector$\overrightarrow {B}$ and at the end of vector $\overrightarrow{B}$you draw a line parallel to vector$\overrightarrow{A}$ (the dashed lines in Figure 1.9 (b)). In this way, you obtain a parallelogram. From the origin of the two vectors, you draw a diagonal of the parallelogram as Repeat activity 1.2 shown in (Figure 1.9 (b)). The diagonal is the resultant $\overrightarrow{R}$ of the two vectors. thus

$\overrightarrow{R}$ = $\overrightarrow{A}$ +$\overrightarrow{B}$

Since vector addition is commutative,

$\overrightarrow{A}$ +$\overrightarrow{B}$ = $\overrightarrow{B}$ +$\overrightarrow{A}$

Polygon method of vector addition

This law is used for the addition of more than two vectors. According to this law, if you have a large number of vectors, place the tail end of each successive vector at the head end of previous one. The resultant of all vectors can be obtained by drawing a vector from the tail end of first to the head end of the last vector.

Figure 1.5. The polygon rule for the addition of vectors

the resultant force must added as figure above ., $\overrightarrow{R}$ = $\overrightarrow{D}$ + $\overrightarrow{A}$ + $\overrightarrow{C}$ + $\overrightarrow{B}$.

Note: To determine the resultant of two vectors acting:in the same direction, add the given vectors and take the common direction.

$\overrightarrow{R}$ = $\overrightarrow{A}$ +$\overrightarrow{B}$

• in opposite directions, get the difference and take the direc-tion of the vector with the greater value.

$\overrightarrow{R}$ = $\overrightarrow{A}$ -$\overrightarrow{B}$

• When the two vectors are perpendicular

If vectors A and B are perpendicular to each other as shown in figure 1
then the magnitude of the resultant vector $\overrightarrow{R}$ is obtained using the

Pythagoras theorem. Hence, the magnitude of the resultant vector is

$\left| R \right|=\sqrt{A^{2}+B^{2}}$

The direction of the resultant vector is obtained using the trigono-metric equation:

$\Theta=tan^{-1}\left( \frac{B}{A} \right)$

Note: You can compare the result you obtain in each of the three
cases with a ruler and protractor. Surely, you will obtain similar
result.

Figure 1,6 Two perpendicular vectors A and B; and its resultant vector R.

Example 1.1
Two vectors have magnitudes of 6 units and 3 units. What is the magnitude
of the resultant vector when the two vectors are (a) in the same direction,
(b) in opposite direction and (c) perpendicular to each other?
Solution:
You are given with two vectors of magnitudes 6 units and 3 units.
(a) When the two vectors are in the same direction,
|R| = (6+3) units = 9 units.
(b) When the two vectors are in the opposite directions,
|R| = (6−3) units = 3 units.
(c) When the two vectors are perpendicular to each other,
$\left| R \right|=\sqrt{A^{2}+B^{2}}$ = =$\sqrt{6^{2}+3^{2}}$ = 6.7 units.
Please compare the result you obtained here with the value you obtained
with the direct measurement by a ruler.