Introduction
We know that all the physical quantities are classified into two categories depending on the direction and magnitude. The quantities which have both magnitude and direction are known as the vectors and at the same time, the quantities that have the only magnitude are known as a scalar. The vector quantities are additive in nature. If two vectors are in either the opposite direction or the same direction, then we can add or subtract two vectors depending upon the direction directly. If two vectors are in the same direction, then we cannot add them directly, thus we use the resultant vector formula.
In this article, we will learn what is a resultant vector formula, the resultant of two vectors formulas along with solved examples.
What is a Resultant Vector?
Now, let us understand what do we mean by a resultant vector and the importance of the resultant of two vectors formula. So, a resultant vector as its name suggests is the result of two vectors as a consequence of addition. To be more precise, a resultant vector is nothing but a combination, in simpler words, a resultant vector can be described as the sum of two or more vectors which are having different magnitude and direction.
Thus we say that the resultant vector is a new vector that results in the combined effect of all the vectors incorporated in the process of addition. Whenever we add two or more two vectors, the consequence of that addition will be the resultant vector.
Resultant of Two Vectors Formula
Now, let us have a look at the resultant of the two vectors formula in detail followed by a small derivation. The definition of resultant of two vectors states that if two vectors acting simultaneously can be represented in magnitude and direction by the two adjacent sides of a parallelogram formed from a point then their resultant is expressed in magnitude and direction is given by the diagonal of the parallelogram drawn from the same point.
(Image will be uploaded soon)
Consider two vectors A and b and these two vectors are represented in magnitude and direction by two adjacent sides OP and OS of the parallelogram OPQS. Let \[\theta\] be the angle made by the two vectors. Now, according to the law of parallelogram, the diagonal of the parallelogram will be the resultant vector. Let Rbe the resultant vector of these two vectors, then the resultant of two vectors formula is given by the following expression:
\[\Rightarrow R = \sqrt{A^{2}+B^{2} + 2AB Cos \theta} \]
Where,
\[A \] and \[B\] - The magnitude of the two vectors acting in the same directions
\[\theta\] - The angle between the two vectors
Case 1:
When two vectors are acting in the same direction, then we know that angle made by the two vectors will be \[0^{\circ}\]. Therefore, the resultant vector is given by:
\[\Rightarrow R = \sqrt{A^{2} + B^{2} + 2AB Cos 0} \]
\[\Rightarrow R = \sqrt{A^{2} + B^{2} + 2AB} \]
\[ \Rightarrow R = \sqrt{(A+B)^{2}} = A+B\]
Thus, when two vectors acting in the same direction then the resultant vector will be \[ R = A+B \].
Case 2:
When two vectors are acting in the opposite direction, then we know that angle made by the two vectors will be \[180^{\circ}\] Therefore, the resultant vector is given by:
\[\Rightarrow R = \sqrt{A^{2} +B^{2} + 2AB Cos 180} \]
\[\Rightarrow R = \sqrt{A^{2}+B^{2} - 2AB} \]
\[\Rightarrow R = \sqrt{(A-B)^{2} }= A-B \]
Thus, when two vectors acting in the opposite direction then the resultant vector will be \[R = A-B \].
Case 3:
When two vectors are at the right angle to each other, then we know that angle made by the two vectors will be \[90^{\circ}\] Therefore, the resultant vector is given by:
\[ \Rightarrow R = \sqrt{A^{2}+B^{2}+2AB Cos 90} \]
\[\Rightarrow R = \sqrt{A^{2}+B^{2}} \]
\[\Rightarrow R = \sqrt{A^{2}+B^{2}} \]
Thus, when two vectors acting in the same direction then the resultant vector will be \[\Rightarrow R = \sqrt{A^{2}+B^{2}} \].
The resultant vectors are important while studying the resolution of vectors such as the force vector, velocity vector, etc.
Examples:
Two force vectors of magnitude 6N each are inclined at \[60^{\circ}\] with each other and acting on the body. Calculate the resultant of these two forces.
Sol:
Given,
The magnitude of vector 1 (say) = A = 6 N
The magnitude of vector 2 (say) = B = 6 N
The angle between the vector 1 and vector 2 = \[\theta = 60^{\circ}\]
Now, we are asked to determine the resultant vectors of the given two force vectors. We know that the resultant vector formula is given by the following expression:
\[\Rightarrow R = \sqrt{A^{2}+B^{2}+2AB Cos \theta} \].....(1)
Where,
\[A \] and \[B\] - The magnitude of the two vectors acting in the same directions
\[\theta\] -The angle between the two vectors
Substituting the values of the A, B and \[\theta\] in the equation (1) and simplify we get:
\[\Rightarrow R = \sqrt{6^{2}+6^{2}+2(6)(6) Cos 60} \]
\[\Rightarrow R = \sqrt{36+36+2(36) \frac{1}{2}} \]
\[\Rightarrow R = \sqrt{36+36+36} \]
\[\Rightarrow R = 10.39 \]
Therefore, the resultant vectors of the given two force vectors are 10.39.
Two vectors of magnitude 6N and 9N each are inclined at \[45^{\circ}\] with each other and acting on the body. Calculate the resultant of these two forces.
Sol:
Given,
The magnitude of vector 1 (say) = A = 6 N
The magnitude of vector 2 (say) = A = 9 N
The angle between the vector 1 and vector 2 = \[\theta = 45^{\circ}\]
Now, we are asked to determine the resultant vectors of the given two force vectors. We know that the resultant of two vectors formula is given by the following expression:
\[\Rightarrow R = \sqrt{A^{2}+B^{2} + 2ABCos \theta} \].....(1)
Where,
\[A\] and \[B\] - The magnitude of the two vectors acting in the same directions
\[\theta\] -The angle between the two vectors
Substituting the values of the A, B and \[\theta\] in the equation (1) and simplify we get:
\[\Rightarrow R = \sqrt{6^{2} + 9^{2}+2(6)(9) Cos 45} \]
\[\Rightarrow R = \sqrt{36+81+2(54) \frac{1}{\sqrt{2}} } \]
\[\Rightarrow R = \sqrt{193.367} \]
\[\Rightarrow R = 13.905 \]
Therefore, the resultant vectors of the given two force vectors is \[13.905 \].
FAQs on Resultant Vector Formula
1. How do we calculate the resultant force of two vectors?
Two forces that are acting in opposite directions will produce a resultant force that is smaller than each individual force considered. To determine the resultant force we should subtract the magnitude of the smaller force from the magnitude of the larger force. The direction of the resultant force will be in the same direction as the larger force among the two.
2. What is the resultant vector definition?
The resultant of two vectors is defined as when two or more vectors acting simultaneously then, they can be represented in magnitude and direction by the two adjacent sides of a parallelogram formed by a point, then their resultant is written in the magnitude and direction of the resultant vector is identified by the diagonal of the parallelogram drawn from the same point.
3. What is the importance of the resultant vector formula and how to find resultant vector?
The formula of resultant vectors play a major role in mechanics. In order to determine the value of certain physical quantities, we use the resultant vector formula. The resultant formula is given by:
\[\Rightarrow R = \sqrt{A^{2}+B^{2}+2AB Cos \theta} \]
Where,
\[A \] and \[B\] - The magnitude of the two vectors acting in the same directions
\[\theta\] - The angle between the two vectors
