Vector is a physical quantity that has both magnitude and direction such as velocity, displacement, force, etc. We can say that a vector needs both magnitude and direction for its complete description. For example, a velocity of 50 km/hr, north. Here, an object is moving at a speed of 50 km/hr (magnitude) toward the north (direction).

A variety of mathematical operations can be performed with and upon vectors. One such vector operation is the addition of vectors. The addition of two vector quantities cannot be done by using ordinary algebra.

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In this mathematics article, we will study vector addition and different types of laws of vector addition, applications of vector addition, properties of vector addition, and also solve some problems on laws of vector addition that will help you to understand the topic easily.

Addition of vectors is the operation to add two or more vectors together to form a vector sum. A vector quantity is denoted by an arrow over a letter, or with an alphabet written in bold, or a line segment with an arrow at one end, where the arrow tells the direction of the vector.

Two or more vectors can be equal if they have the same magnitude and direction. When we multiply the vector quantity with a positive number, then its magnitude changes but its direction remains unchanged. However, when we multiply it with a negative integer, both magnitude and direction change.

There are some conditions that need to be followed while doing vector addition and these are as follows:

• Condition 1: Two or more vectors can be added only if they have the same nature.
• Condition 2: We cannot add a vector quantity with the scalar quantity.

Laws of Vector Addition

There are 3 laws of vector addition that are as follows:

• 1st Law: Triangle law of Vector Addition
• 2nd Law: Parallelogram law of Vector Addition
• 3rd Law: Polygon law of Vector Addition

1. Triangle Law of Vector Addition

Triangle law of vector addition states that if two vectors are represented in magnitude and direction by two sides of a triangle taken in the same order, then their resultant is represented by the third side of the triangle taken in the opposite order.

Triangle law of vector addition formula

Derivation of triangle law of vector addition

Let ∠ QPE = θ.

Then, in the right-angled triangle OEQ, we have

2. Parallelogram Law of Vector Addition

Parallelogram law of vector addition states if two vectors act along two adjacent sides of a parallelogram (with a 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.

Parallelogram law of vector addition formula

Derivation of parallelogram law of vector addition

3. Polygon Law of Vector Addition

The polygon law of vector addition states that if the sides of a polygon are taken in the same order to represent a number of vectors in magnitude and direction, then the resultant vector can be represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

Polygon law of vector addition formula

Derivation of polygon law of vector addition

Consider there being a polygon of 5-side.

In triangle ABC,

By triangle law of vector addition, we have

AB + BC = AC ……….(i)

And, similarly

AC + CD = AD ………..(ii)

and, AD + DE = AE ……(iii)

Similarly, we will do this for all sides in a polygon (here we have only 5-sided).

Then, adding equations (i), (ii) and (iii), we get

AB + BC + CD + DE = AE.

Here, AE = resultant vector and others are normal vector sides.

Note: For n-sides, it can be written analogously and stated as the above theorem.

Vector Addition using Components

The most common way of adding vectors is with components. Each entry in the two-dimensional ordered pair (a, b) or three-dimensional triplet (a, b, c) is called the component of the vector. The entries correspond to the number of units the vector has in the x, y and (for three-dimensional case) z directions of a space or plane. In other words, components are simply the coordinates of the point associated with a vector.

Parallelogram Law of Vector Addition Procedure

The Parallelogram Law is a method used to find the resultant of two vectors acting from the same point.

Follow these simple steps:

Step 1:
Draw the first vector using an appropriate scale (like 1 cm = 10 N), and point it in its given direction.

Step 2:
From the same starting point (tail), draw the second vector using the same scale and direction as given.

Step 3:
Now treat these two vectors as adjacent sides of a parallelogram. Use a ruler to complete the opposite sides of the parallelogram.

Step 4:
Draw a diagonal from the common starting point (where both vectors begin) to the opposite corner of the parallelogram.
This diagonal represents the resultant vector—showing both its magnitude and direction.

This method gives a clear visual way to add two vectors and understand how their directions and magnitudes combine to form a single, equivalent vector.

Important Properties of Vector Addition

Important properties of vector addition are given below:

Property 1: Vector Addition is Commutative: If two vectors \(\vec{a}\) and \(\vec{b}\) are added together, then
\(\vec{a}+\vec{b}=\vec{b}+\vec{a}\), this means that the order of vectors does not change the result of the addition.

Property 2: Vector Addition is Associative: If there are three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\), then

\((\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})\), this mutual grouping of vectors has no effect on the result when adding three or more vectors together.

Property 3: Vector Addition is Distributive: If there are two vectors \(\vec{a}\), \(\vec{b}\) and one scaler ‘m’, then

\(m(\vec{a}+\vec{b})=m\vec{a}+m\vec{b}\), this means that the sum of the scalar times the sum of two vectors equals the sum of the scalar times of the two vectors separately.

Property 4: Existence of Identity: For any vector \(\vec{a}\),

\(\vec{a}+\vec{0}=\vec{a}\). Here, 0 is the additive identity.

Property 5: Existence of Inverse: For any vector \(\vec{a}\),

\(\vec{a}+(-\vec{a})=\vec{0}\). Here, an additive inverse exists for every vector.

Important Properties of Vector Subtraction 

Subtracting one vector from another is similar to adding vectors, but with a small twist.

Let’s say you want to subtract vector b from vector a (written as a – b).
Instead of subtracting directly, you can add vector a to the opposite of vector b.

This means:

a – b = a + (–b)

Here, –b means vector b with its direction reversed. The length (magnitude) stays the same, but it points in the opposite direction.

Example:

If vector b points to the right, then –b points to the left.

So, to subtract vectors:

1. Flip the direction of the vector you are subtracting (make it negative).
    
2. Add it to the first vector using the usual head-to-tail method or parallelogram method.

Laws of Vector Addition Examples

Example 1: If the position vectors of the points A(2, 5), B(3, -4) and C(7, -2) are \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) respectively, then compute the value \(\vec{a}+2\vec{b}-3\vec{c}\).

Solution: Given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are the position vectors of the points A(2, 5) , B(3, -4) and C(7, -2).

Then, \(\vec{a}=2\hat{i}+5\hat{j}\), \(\vec{b}=3\hat{i}-4\hat{j}\) and \(\vec{c}=7\hat{i}-2\hat{j}\).

Therefore,

\(\vec{a}+2\vec{b}-3\vec{c}=(2\hat{i}+5\hat{j})+2(3\hat{i}-4\hat{j})-3(7\hat{i}-2\hat{j})\)

\(\vec{a}+2\vec{b}-3\vec{c}=(2+6-21)\hat{i}-(5-8+6)\hat{j}\)

\(\vec{a}+2\vec{b}-3\vec{c}=-13\hat{i}+3\hat{j}\).

Example 2: Two vectors \(\vec{A}\) and \(\vec{B}\) of magnitude 5 units and 7 units respectively make an angle \(60^{\circ}\) with each other. Find the magnitude of the resultant vector and its direction with respect to the \(\vec{A}\).

Solution: By using the triangle law of vector addition, we have

The magnitude of the resultant vector \(\vec{R}\) is given by

\(R = \sqrt{(A^{2}+B^{2}+2ABcos\theta)}\)

\(R = \sqrt{(5^{2}+7^{2}+2*5*7cos60^{\circ})}\)

\(R = \sqrt{(25+49+35)} = \sqrt{109}\).

The angle \(\alpha\) between \(\vec{R}\) and \(\vec{A}\) is given by

\(\tan\alpha=\frac{B sin\theta}{A+B cos\theta}\)

\(\tan\alpha=\frac{7sin60^{\circ}}{5+7cos60^{\circ}}\cong\frac{7\sqrt{3}}{17}\cong0.713\)

\(\therefore\) \(\alpha \cong 35^{\circ}\).

Example 3:

Consider two vectors, A and B, given by their components:

A = (3, -2)

B = (-1, 4)

• a) Find the vector sum A + B.
• b) Find the magnitude of the resultant vector.

Solution:

• a) To find the vector sum, we add the corresponding components of the vectors:

A + B = (3, -2) + (-1, 4) = (3 + (-1), -2 + 4) = (2, 2)

• b) The magnitude of the resultant vector can be found using the Pythagorean theorem:

|A + B| = sqrt((2)^2 + (2)^2) = sqrt(8) ≈ 2.83

Example 4:

Suppose you are navigating a boat on a river. The boat is moving with a velocity of 10 m/s northward, while the river current is flowing with a velocity of 5 m/s eastward. Determine the resulting velocity of the boat.

Solution:

To determine the resulting velocity, we can represent the velocities as vectors and apply vector addition.

Let A represent the velocity of the boat (10 m/s northward) and B represent the velocity of the river current (5 m/s eastward).

• a) Represent the velocities as vectors:

A = (0, 10) m/s (northward)

B = (5, 0) m/s (eastward)

• b) Find the vector sum A + B:

A + B = (0, 10) + (5, 0) = (0 + 5, 10 + 0) = (5, 10)

• c) The resulting velocity of the boat is given by the magnitude and direction of the resultant vector:

|A + B| = sqrt((5)^2 + (10)^2) = sqrt(125) ≈ 11.18 m/s

The direction can be determined using trigonometry:

tanθ = 10/5⇒θ≈63.43∘

Therefore, the resulting velocity of the boat is approximately 11.18 m/s at an angle of approximately 63.43 degrees north of east.

Example 5:

Consider two vectors A = 3i + 2j and B = -i + 4j. We will calculate the vector sum using the parallelogram law.

Solution:

Using the parallelogram law, the vector sum C = A + B can be calculated as:

C = A + B = (3i + 2j) + (-i + 4j) = 3i - i + 2j + 4j = 2i + 6j

Hence, the vector sum C is 2i + 6j.

Example 6:

Suppose a boat is sailing in a river with a velocity of 4 m/s to the east, while the river current has a velocity of 2 m/s to the south. Find the resultant velocity of the boat.

Solution:

Let's represent the velocity of the boat as vector A = 4i (eastward) and the velocity of the river current as vector B = -2j (southward).

The resultant velocity of the boat can be found by adding vectors A and B:

C = A + B = 4i + (-2j) = 4i - 2j

Hence, the resultant velocity of the boat is 4i - 2j.

Example 7:

Suppose a person walks 5 km due north, then turns and walks 12 km due east. Find the resultant displacement of the person.

Solution:

Let's represent the displacement due north as vector A = 5j and the displacement due east as vector B = 12i.

The resultant displacement of the person can be found by adding vectors A and B:

C = A + B = 5j + 12i

Hence, the resultant displacement of the person is 5j + 12i.

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