Addition of Vectors: It is significantly necessary for students to understand the properties of vectors before they engage in executing any mathematical operation with them. Vectors can be used to perform a wide range of mathematical operations, addition is one such operation. The result of addition of vectors can be determined simply by adding two vectors (or resultant). This procedure of adding two or more vectors is more challenging than scalar addition.

The total distance travelled, in this case, is \(20\) miles. However, there is no displacement. Each of the North and South displacements is a vector quantity, and the opposite directions cause the individual displacements. Let us explore the addition of vectors and their properties with solved examples in detail.

Addition of Vectors: Details

Vectors are written with an alphabet and an arrow over them and are represented as a combination of direction and magnitude.

The operation to add two or more vectors together to form a vector sum is known as the addition of vectors. The addition of vectors is done in two ways, either through triangle law or parallelogram law.
If two vectors have the same direction, the sum of their magnitudes in the same direction is equal to the sum of their directions.

If the two vectors are in opposite directions, the resultant of the vectors is the magnitude difference between the two vectors and is in the direction of the larger vector. Using vector addition, two vectors, \(\overrightarrow x \) and \(\overrightarrow y, \) can be added together, and the resultant vector can be expressed as \(\overrightarrow R = \overrightarrow x + \overrightarrow y .\)

Before we can learn about the properties of vector addition in maths, we must first understand the requirements that must be met while adding vectors.

The following are the requirements:
1. Only vectors of the same type can be combined together. Acceleration, for example, should be added with only acceleration and not displacement.
2. We can’t add vectors with any scalars. i.e. we cannot add \(2\) with vector \(\overrightarrow a .\)

Consider two vector \(\overrightarrow a \) and \(\overrightarrow b \) where, \(\overrightarrow a = {a_1}i + {a_2}j + {a_3}k\) and \(\overrightarrow b = {b_1}i + {b_2}j + {b_3}k.\) Then the resultant vector \(\overrightarrow R = \overrightarrow a + \overrightarrow b = \left( {{a_1} + {b_1}} \right)i + \left( {{a_2} + {b_2}} \right)j + \left( {{a_3} + {b_3}} \right)k.\)

Addition of Vectors: Properties

The addition of vectors differs from the addition of algebraic numbers. Here are some of the most significant properties to think about when adding vectors:

1. Vector addition is commutative: It means the order of vectors does not affect the result of the addition. If two vectors \(\overrightarrow a \) and \(\overrightarrow b \) are added together, then \(\overrightarrow a + \overrightarrow b = \overrightarrow b + \overrightarrow a \)

2. Vector addition is associative: The mutual grouping of vectors has no effect on the result when adding three or more vectors together.
\(\left( {\overrightarrow a + \overrightarrow b } \right) + \overrightarrow c = \overrightarrow a + \left( {\overrightarrow b + \overrightarrow c } \right)\)

3. Vector addition is distributive: It indicates that the sum of scalar times the sum of two vectors equals the sum of the scalar times of the two vectors separately.
\(m\left( {\overrightarrow a + \overrightarrow b } \right) = m\overrightarrow a + m\overrightarrow b \)

4. Existence of Identity: For any vector \(\overrightarrow a ,\,\overrightarrow a + \overrightarrow 0 = \overrightarrow a \)
Here, \(\overrightarrow 0 \) is the additive identity.

5. Existence of inverse: For any vector \(\overrightarrow a ,\,\overrightarrow a + \left( { – \overrightarrow a } \right) = \overrightarrow 0 \)
So, an additive inverse exists for every vector.

Addition of Vectors: Formulas & Laws

The addition of vectors is done in these two ways:

1. Triangle law of vector addition: The triangle law of vector addition states that when two vectors are represented as two sides of a triangle with the same order of magnitude and direction, the magnitude and direction of the resultant vector is represented by the third side of the triangle.

\(AB = \overrightarrow a \) and \(BC = \overrightarrow b ,\) then resultant line \(AC = \overrightarrow a + \overrightarrow b \)
If \(a = \) magnitude of \(\overrightarrow a \)
\(b = \) magnitude of \(\overrightarrow b \)
\(\theta = \) angle between \(\overrightarrow a \) and \(\overrightarrow b \)
Then, the magnitude of \(\overrightarrow a + \overrightarrow b \) is calculated by the formula \(\sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\theta } \)

2. Parallelogram law of vector addition: Parallelogram law of vector addition states that If two vectors act along two adjacent sides of a parallelogram (with magnitude equal to the length of the sides) both pointing away from the common vertex, the resultant is represented by the diagonal of the parallelogram passing through the same common vertex.

If \(\overrightarrow a \) and \(\overrightarrow b \) represent the adjacent sides of the parallelogram \(ABCD\) as shown in the figure, then the result is the diagonal \(AC\) of the parallelogram \(ABCD\) which passes through the same common vertex.

Hence, we can deduce that the triangle laws of vector addition and the parallelogram laws of vector addition are equivalent.

Solved Examples – Addition of Vectors

Let us understand the concept of the addition of vectors examples with solutions.

Q.1. If the position vectors of the points \(A\left( {3,\,4} \right),\,B\left( {5,\, – 6} \right)\) and \(C\left( {4,\, – 1} \right)\) are \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \) respectively, compute \(\overrightarrow a + 2\overrightarrow b – 3\overrightarrow c .\)
Ans: Let \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \) are the position vectors of the points \(A\left( {3,\,4} \right),\,B\left( {5,\, – 6} \right)\) and \(C\left( {4,\, – 1} \right).\)
Then, \(\overrightarrow a = 3\widehat i + 4\widehat j,\,\overrightarrow b = 5\widehat i – 6\widehat j\) and \(\overrightarrow c = 4\widehat i – \widehat j\)
Therefore,\(\overrightarrow a + 2\overrightarrow b – 3\overrightarrow c = 3\widehat i + 4\widehat j + 2\left( {5\widehat i – 6\widehat j} \right) – 3\left({4\widehat i  – \widehat j }\right)\)
\( = 3 \widehat i + 4\widehat j + 10\widehat i – 12\widehat j – 12\widehat i + 3\widehat j\)
\( = \widehat i – 5\widehat j\)

Q.2. Find the magnitude of the sum of a \(15\,{\rm{km}}\) displacement and a \(25\,{\rm{km}}\) displacement when the angle between them is \({60^{\rm{o}}}\)
Ans: Here, \(a=\) magnitude of \(\overrightarrow a = 15\)
\(b=\) magnitude of \(\overrightarrow b = 25\)
\(\theta = \) Angle between \(\overrightarrow a \) and \(\overrightarrow b = {60^{\rm{o}}}\)
Hence, Then, the magnitude of \(\overrightarrow a + \overrightarrow b \) which is the resultant sum will be \( = \sqrt {{a^2} + {b^2} + 2ab\,\cos \,\theta } \)
\( = \sqrt {{{15}^2} + {{25}^2} + 2 \times 15 \times 25\,\cos \,{{60}^{\rm{o}}}} \)
\( = \sqrt {225 + 625 + \frac{{750}}{2}} \)
\( = \sqrt {850 + 375} \)
\( = \sqrt {1225} \)
\(=35.\)

Q.3. If \(D\) is the mid-point of side \(BC\) of a triangle \(ABC\) such that \(\overrightarrow {AB} + \overrightarrow {AC} = \overrightarrow {\lambda AD} ,\) write the value of \(\lambda .\)
Ans: Given \(D\) is the midpoint of the side \(BC\) of a triangle \(ABC\) such that \(\overrightarrow {AB} + \overrightarrow {AC} = \overrightarrow {\lambda AD} .\)
Let \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \) are the position vectors of \(AB, BC\) and \(CA.\)
Now, the position vector of \(D\) is \(\frac{{\overrightarrow b + \overrightarrow c }}{2}.\)
Then, \(\overrightarrow {AB} = \overrightarrow b – \overrightarrow a \) and \(\overrightarrow {AC} = \overrightarrow c – \overrightarrow a \)
\(\overrightarrow {AD} = \frac{{\overrightarrow b + \overrightarrow c }}{2} \,- \overrightarrow a \)
Now, we have,
\(\overrightarrow {AB} + \overrightarrow {AC} = \overrightarrow {\lambda AD} ,\)
\( \Rightarrow \overrightarrow b – \overrightarrow a + \overrightarrow c – \overrightarrow a = \lambda \left( {\frac{{\overrightarrow b + \overrightarrow c }}{2} – \overrightarrow a } \right)\)
\( \Rightarrow \overrightarrow b + \overrightarrow c – 2\overrightarrow a = \lambda \left( {\frac{{\overrightarrow b + \overrightarrow c – 2\overrightarrow a }}{2}} \right)\)
\( \Rightarrow \lambda = 2\)

Q.4. If \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \) are position vectors of the vertices \(A, B\) and \(C\) respectively, of a triangle \(ABC,\) write the value of \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CA.} \)
Ans: Given \(\overrightarrow a , \overrightarrow b , \overrightarrow c \) are the position vectors of \(A, B\) and \(C\) respectively.
Then, \(\overrightarrow {AB} = \overrightarrow b – \overrightarrow a \)
\(\overrightarrow {BC} = \overrightarrow c – \overrightarrow b \)
\(\overrightarrow {CA} = \overrightarrow a – \overrightarrow c \)
Consider, \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CA} = \overrightarrow b – \overrightarrow a + \overrightarrow c – \overrightarrow b + \overrightarrow a – \overrightarrow c \)
\( = \overrightarrow 0 \)

Q.5. Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.
Ans: Let \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \) are the position vectors of the vertices \(A, B\) and \(C\) respectively.
Then we know that the position vector of the centroid \(O\) of the triangle is \(\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}.\)
Therefore, sum of the three vectors \(\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} = \overrightarrow a – \left( {\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}} \right) + \overrightarrow b – \left( {\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}} \right) + \overrightarrow c – \left( {\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}} \right)\)
\( = \left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) – 3\left( {\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}} \right) = \overrightarrow 0 \)
Hence, the sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.

Summary

In this article, we learnt about the addition of vectors, their properties and examples. The procedure of adding two or more vectors is different from scalar addition. Only vectors of the same type can be combined together. We can’t add vectors with scalars. There are two ways for the addition of vectors, and they are Triangle law and Parallelogram law of vector addition which is equivalent.

Frequently Asked Questions (FAQs) – Addition of Vectors

Frequently asked questions related to addition of vectors is listed as follows:

Q.1. Explain the addition of vectors.
Ans: The operation of adding two or more vectors together to form a vector sum is known as the addition of vectors. The addition of vectors is done through these two ways, either through Triangle law or Parallelogram law. When two vectors are placed head to tail, the vector sum is determined by drawing the vector from the tail to the head.

Q.2. How to implement the addition of vectors?
Ans: If two vectors have the same direction, the sum of their magnitudes in the same direction is equal to the sum of their directions. If the two vectors are in opposing directions, the resultant of the vectors is the magnitude difference between the two vectors and is in the direction of the larger vector. In all other cases, we use the concept of either triangle law or parallelogram law for the addition of vectors.

Q.3. What are examples of the addition of vectors?
Ans: Consider two vector \({\overrightarrow a }\) and \({\overrightarrow b }\) where, \(\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k\) and \(\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k.\) Then the resultant vector \(\overrightarrow R = \overrightarrow a + \overrightarrow b = \left( {{a_1} + {b_1}} \right)\widehat i + \left( {{a_2} + {b_2}} \right)\widehat j + \left( {{a_3} + {b_3}} \right)\widehat k.\)
For example:
1. if \(\overrightarrow a = \widehat i + 2\widehat j + 3\widehat k\) and \(\overrightarrow b = 4\widehat i + 5\widehat j + 6\widehat k.\) Then the resultant vector \(\overrightarrow R = \overrightarrow a + \overrightarrow b = 5\widehat i + 7\widehat j + 9\widehat k.\)
2. If \(\overrightarrow a + \overrightarrow b + \overrightarrow c \) are the position vectors of the points \(A\left( {3,\,4} \right)\,B\left( {5,\, – 6} \right)\) and \(C\left( {4,\, – 1} \right),\) respectively. Now suppose we have to find out \(\overrightarrow a + 2\overrightarrow b – 3\overrightarrow c \)
Then, \(\overrightarrow a = 3\widehat i + 4\widehat j,\,\overrightarrow b = 5\widehat i – 6\widehat j\) and \(\overrightarrow c = 4\widehat i – \widehat j\)
Therefore, \(\overrightarrow a + 2\overrightarrow b – 3\overrightarrow c = 3\widehat i + 4\widehat j + 2\left( {5\widehat i – 6\widehat j} \right) – 3\left( {4\widehat i – \widehat j} \right)\)
\( = 3\widehat i + 4\widehat j + 10\widehat i – 12\widehat j – 12\widehat i + 3\widehat j\)
\( = \widehat i – 5\widehat j\)

Q.4. What are the uses of the addition of vectors?
Ans: The addition of vectors plays an important role in engineering, which involves forces, electric fields, magnetic fields, momentum, angular momentum, position, trajectories, polarization, current density, magnetization, velocities, torque etc. Sum of any two vectors can be done geometrically by building two sides of a triangle with the given vectors, with the resultant vector on the third side. Because these are fundamental mathematical laws, they hold true for all vectors, including vectors from physics, and are thus frequently employed in engineering.

Q.5. Can the sum of two vectors have a zero?
Ans: The sum of two vectors can only be \(0\) if they are in the opposite direction and have the same magnitude. Other than this, the sum of any two vectors can’t be zero.