The laws of Physics Topics are used to explain everything from the smallest subatomic particles to the largest galaxies.
Some Special Cases of Addition of Two Vectors
1. Two parallel vectors (α = 0) : In this case, sinα = 0 and cosα = 1. Thus from equation (1), we can write,
c = \(\sqrt{a^2+b^2+2 a b}\) = \(\sqrt{(a+b)^2}\) = (a+ b)
From equation (2), we can conclude that tanθ = 0, or, θ = 0°.
Hence, magnitude of the resultant is the sum of the magnitudes of the vectors and it is directed along the vectors. It is simply a scalar addition.
2. Two anti-parallel vectors (α = 180°): In this case, sinα = 0 and cosα = -1. Therefore, from equation (1), we get,
c = \(\sqrt{a^2+b^2-2 a b}\) = \(\sqrt{(a-b)^2}\) or, c = |a – b| and from equation (2), tan θ = 0 or, θ = 0° when a > b or θ = 180° when b > a.
Hence, magnitude of the resultant is the diffërence in the magnitudes of the vectors and its direction is along the larger vector. It is also a simple scalar operation.
3. Two equal and anti-parallel vectors (a = b and α = 180°): In this case sinα = 0 and cosα = -1.
Hence, using equation (1)
c = \(\sqrt{a^2+a^2+2 a^2 \cos 180^{\circ}}\) = \(\sqrt{2 a^2-2 a^2}\) = 0
Therefore, magnitude of the resultant is zero. It is essentially a null vector (see section 2.6).
4. Two orthogonal (perpendicular to each other vectors (α = 90°): In this case, sinα = 1 and cosα = 0
∴ From equation (1) and (2),
c = \(\sqrt{a^2+b^2+2 a b \cos 90^{\circ}}\) = \(\sqrt{a^2+b^2}\)
and tanθ = \(\frac{b}{a}\)
Therefore, we can conclude that,
- maximum possible value of the resultant = sum of magnitudes of the vectors i.e., c (max) = a + b
- minimum possible value of the resultant = difference of magnitudes of the vectórs i.e.. c(min) = |a – b|
- if the vectors are equal in magnitude and anti-parallel, then the resultant will be 0.
Properties of Vector Addition
Commutative rule:
Vector addition is commutative. If \(\vec{a}\) and \(\vec{b}\) are two vectors, then by this rule, \(\vec{a}\) + \(\vec{b}\) = \(\vec{b}\) + \(\vec{a}\).
This means that vectors can be added in any order.
Proof:
With reference to Fig., OA and OC representing two vectors are two adjacent arms of the parallelogram OABC.
Here, \(\vec{a}\) = \(\overrightarrow{O A}\) = \(\overrightarrow{C B}\) and \(\vec{b}\) = \(\overrightarrow{O C}\) = \(\overrightarrow{A B}\)
So, \(\vec{a}\) + \(\vec{b}\) = \(\overrightarrow{O A}\) + \(\overrightarrow{A B}\) = \(\overrightarrow{O B}\) = \(\vec{c}\), (as per the mangle law of addition of vectors).
Again, \(\vec{b}\) + \(\vec{a}\) = \(\overrightarrow{O C}\) + \(\overrightarrow{C B}\) = \(\overrightarrow{O B}\) = \(\vec{c}\)
Hence, \(\vec{a}\) + \(\vec{b}\) = \(\vec{b}\) + \(\vec{a}\) or in other words, vector sum is commutative.
Associative rule:
Vector addition is associative. To add any three vectors, addition may be initiated with any two of the vectors. Mathematically, (\(\vec{a}\) + \(\vec{b}\)) + \(\vec{c}\) = \(\vec{a}\) + (\(\vec{b}\) + \(\vec{c}\)), where \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are the given vectors. This rule holds good for the addition of more than three vectors as well.
Proof:
Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be represented by the three arms OA, AB and BC of the quadrilateral OABC.
As shown in Fig.(a),
\(\vec{a}\) + \(\vec{b}\) = \(\overrightarrow{O A}\) + \(\overrightarrow{A B}\) = \(\overrightarrow{O B}\)
and \(\overrightarrow{O B}\) + \(\overrightarrow{B C}\) = (\(\vec{a}\) + \(\vec{b}\)) + \(\vec{c}\) = \(\overrightarrow{O C}\)
Again, as per Fig.(b),
\(\vec{b}\) + \(\vec{c}\) = \(\overrightarrow{A B}\) + \(\overrightarrow{B C}\) = \(\overrightarrow{A C}\)
∴ \(\vec{a}\) + (\(\vec{b}\) + \(\vec{c}\)) = \(\overrightarrow{O A}\) + \(\overrightarrow{A C}\) = \(\overrightarrow{O C}\)
Hence, \(\vec{a}\) + (\(\vec{b}\) + \(\vec{c}\)) = (\(\vec{a}\) + \(\vec{b}\)) + \(\vec{c}\)
It is to be noted that scalar addition also follows these two rules.