Combination of Vectors Physics Notes
Graphical Method:
The following facts should be followed while combining the vectors:
- When a vector is displaced at any point within the parallel space without changing its magnitude then, the vector remains unchanged.
- Addition of two vectors always give a vector. It is called the resultant vector.
- Only those vectors that represent similar physical quantities can be added.
The laws of addition of vectors are:
(a) Triangle Law of Vector Addition
(b) Parallelogram law of Vector Addition
(c) Polygon law of Vector Addition
(a) Triangle Law of Vector Addition:
It states that, “If two vectors acting on a body are represented both in magnitude and direction by two sides of a triangle taken out in an same order then the resultant is represented by the third side of the triangle taken in the opposite order in magnitude and direction.”
This simply means that, if you have two vectors that represents the two sides of the triangle then the third side of that triangle will represent their resultant.
Figure (a) shows the two vectors \(\vec{P}\) and \(\vec{Q}\). To find out their resultant (sum) by using the triangle law of vector addition, draw the vector \(\vec{AB}\) = \(\vec{P}\). Now move the vector \(\vec{Q}\) parallel to itself, such that its tail coincides with the tip of \(\vec{P}\) vector. Mark the tip of vector \(\vec{Q}\) as C [figure (b)]. Thus the vector \(\vec{P}\) and \(\vec{Q}\) have been represented by the two sides \(\vec{AB}\) and \(\vec{BC}\) or a triangle taken is the same order.
Therefore according to the law of vector addition, the vector \(\vec{AC}\) = \(\vec{R}\) (say) drawn from the tail of the vector P to the tip of the vector Q is the third side of the triangle taken in the opposite order and hence it represents the sum or the resultant of the vectors \(\vec{P}\) and Q.
Thus, \(\vec{P}+\vec{Q}=\vec{R}\)
(b) Parallelogram law of Vector Addition:
The parallelogram law is a slightly more explanation of the triangular law.
It states that if “the two vectors are considered to be the adjacent sides of a parallelogram, then, the resultant of the two vectors is given by the vector which is a diagonal passing through the point of contact of the two vectors.”
Let us find out the resultant of the two vectors \(\vec{P}\) and \(\vec{Q}\) as shown in figure (a). To find out the resultant, draw the vector \(\vec{AB}\) = \(\vec{P}\). Now move the vector \(\vec{Q}\) parallel to itself so that its tail coincides with the tail of vector \(\vec{P}\). If we mark its tip as D, then the vector AD represents the vector \(\vec{Q}\). Complete the parallelogram ABCD as shown in figure (b)
Thus, the vector \(\vec{P}\) and \(\vec{Q}\) have been represented by the two adjacent sides \(\vec{AB}\) and \(\vec{AD}\) respectively of a parallelogram. Therefore, according to the parallelogram law of vector addition, the vector \(\vec{AC}\) = \(\vec{R}\) (say), which represents the diagonal of the parallelogram passing through the point A, represents the sum or the resultant of the vector \(\vec{P}\) and \(\vec{Q}\)
Thus \(\vec{P}+\vec{Q}=\vec{R}\)
Note: According to the parallelogram law of addition of vectors \(\overrightarrow{A B}+\overrightarrow{A D}=\overrightarrow{A C}\)
Since, in a parallelogram, the opposite sides are equal and parallel, they must represent equal vectors
∴ \(\overrightarrow{A D}=\overrightarrow{B C}\)
In the above equation, setting \(\overrightarrow{A D}=\overrightarrow{B C}\), we have
\(\overrightarrow{A B}+\overrightarrow{B C}=\overrightarrow{A C}\)
It is the triangle law of vector addition. Hence triangle law of vector addition follows from the parallglogram law of vector addition.
(c) Polygon Law of Vector Addition:
The polygon law is an extension of the earlier two laws of vector addition. It is successive application of the triangle law to more than two vectors. It states that, “If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken out in the same order. Then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.”
Definition 2: Polygon law: If (n-1) sides of a polygon is in sequence. Then the nth side, closing the polygon in the opposite direction, represents the sum of the vectors in both magnitude and direction.
Polygon Law
Let us find out the resultant of the four vectors namely \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) as shown m the figure (A). To fmd out their resultant, draw the vector \(\vec{OP}\) = \(\vec{a}\). Now, move the vector \(\vec{b}\) parallel to itself, so that its tail comcides with the tip of vector \(\vec{a}\). Mark the tip of vector \(\vec{b}\) as Q. Now move the vector \(\vec{c}\) parallel to itself, so that its tail coincides with the tip of the vector \(\vec{b}\) and mark the tip of vector \(\vec{c}\) as S finally move the vector \(\vec{d}\) parallel to itself, so that the tail of vector \(\vec{d}\) coincides with the tip of vector \(\vec{c}\). Then the vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) have been represented by the sides \(\overrightarrow{O P}, \overrightarrow{P Q}, \overrightarrow{Q S}\) and \(\vec{ST}\) taken in the same order figure (b)
According to the polygon law of vector addition, the closing side \(\vec{OT}\) = \(\vec{R}\) (say) is taken in opposite order represents the sum or the resultant of vector \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\)
i.e. \(\vec{a}+\vec{b}+\vec{c}+\vec{d}=\vec{R}\)
Proof: In triangle OPQ, the vectors a and b have been represented by the sides OP and PQ taken in the same order. Therefore, from the triangle law of vector addition, the closing side OQ is taken in the opposite order represents the resultant of vectors a and b. Thus,
