The sum of two vectors can be found using a method called vector addition. One common way to do this is by using the parallelogram law of vector addition. According to this law, if two vectors start from the same point and lie along two sides of a parallelogram, then the diagonal of that parallelogram starting from the same point shows the direction and size (or magnitude) of the resulting vector.

In simpler terms, if you draw both vectors from the same starting point and then complete a parallelogram, the line that cuts across the shape from that starting point is the sum of the two vectors.

Vectors are special quantities in math and physics that have both size and direction—like force or velocity. Just like numbers, vectors can be added, subtracted, and even multiplied using specific rules. The parallelogram method helps us clearly see how two vectors combine into one.

What is Parallelogram Law of Vector Addition?

The Parallelogram Law of Vector Addition helps us find the sum of two vectors that start from the same point. It says that if two vectors are shown as the two sides of a parallelogram starting from one corner, then the diagonal starting from that same corner gives the resultant vector. This diagonal shows both the size (magnitude) and the direction of the combined effect of the two vectors.

In simple words, imagine you draw two arrows (vectors) from the same point to represent two forces or motions. Then, complete a parallelogram using these arrows as sides. The diagonal that starts from the same point where the two arrows begin shows the total or combined vector — which is called the resultant.

This method is useful in physics and math for solving problems where two forces, movements, or directions act at the same time from a single point.

In the image above, we see that OABD is a parallelogram where vectors P and Q are represented through the adjacent sides OA and AD of the parallelogram. Now according to the parallelogram of the vector addition, we get the resultant vector R as the sum of the two vectors, which is represented by the diagonal.

You should note that all the vectors are passing through the same point of origin O, otherwise, it will not satisfy the law.

Therefore we can represent the law in terms of vectors as \( \vec{R}=\vec{P}+\vec{Q} \).

Although this is just the representation and not the formula for the addition of vectors.

Parallelogram Law of Vector Addition Examples

Here are a few examples of the parallelogram of vector addition:

Example 1:
Consider two vectors A and B. Let A have a magnitude of 4 units and point in the north direction (upwards), and let B have a magnitude of 3 units and point in the east direction (rightwards). To find the resultant vector using the parallelogram law, we construct a parallelogram with A and B as its adjacent sides. The diagonal of the parallelogram represents the resultant vector. By measuring the magnitude and direction of the diagonal, we determine the resultant vector.

Example 2:
Let's say we have vector C with a magnitude of 5 units pointing in the northeast direction (45 degrees between north and east). Similarly, we have vector D with a magnitude of 2 units pointing in the southwest direction (45 degrees between south and west). By constructing a parallelogram with C and D as its adjacent sides, the diagonal of the parallelogram represents the resultant vector. Measuring the magnitude and direction of the diagonal gives us the resultant vector.

In both examples, the parallelogram law allows us to determine the magnitude and direction of the resultant vector by constructing a parallelogram using the given vectors as adjacent sides and finding the diagonal of the parallelogram.

Parallelogram Law of Vector Addition Formula

The parallelogram law of vector addition says that if we place two vectors so they have the same initial point, and complete the vectors into a parallelogram, then the sum of the vectors is given by the directed diagonal that starts at the same point as the vectors.

The Parallelogram law of vector addition formula is given below:

If we consider two adjacent vectors P and Q having angle θ between them, and the resultant vector is R which makes an angle of ϕ with the vector P, as shown below.

Formula for Magnitude of resultant R:

\( \begin{array}{l}\left|R\right|=\sqrt{\left(P^2+Q^2+2PQ\cos\theta\right)}\end{array} \)

Formula for Direction of R:

\( \phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right] \)

Parallelogram Law of Vector Addition Proof

The proof of parallelogram law of vector addition is explained below.

Let P and Q be two adjacent vectors at a point represented by two adjacent sides OA and OD of a parallelogram OABD as shown in the figure.

Let θ be the angle between P and Q and R be the resultant vector. Then, according to the parallelogram law of vector addition, diagonal OB represents the resultant of vectors P and Q.

\( \therefore R=P+Q\)

Next, we extend DA to C and draw BC perpendicular to OC.

So from \( \triangle OCB \), we get

\( \begin{array}{l}\ \ \ \ \ \ OB^2=OC^2+BC^2\\\Rightarrow\ OB^2=\left(OA+AC\right)^2+BC^2\ …\left(i\right)\ \ \ \ \ \ \left[\because OC=OA+AC\right]\end{array}\)

In \( \triangle ABC \),

\( \begin{array}{l}\cos\theta=\frac{AC}{AB}\\\Rightarrow AC=AB\cos\theta\\\Rightarrow\ AC=OD\cos\theta=Q\cos\theta\ \ \ \ \ \left[\because AB=OD=Q\right]\end{array}\)

Also,

\( \begin{array}{l}\cos\theta=\frac{BC}{AB}\\\Rightarrow BC=AB\sin\theta\\\Rightarrow\ BC=OD\sin\theta=Q\sin\theta\ \ \ \ \ \left[\because AB=OD=Q\right]\end{array}\)

Thus substituting the values of AC and BC in (i), we get

\( \begin{array}{l}R^2=\left(P+Q\cos\theta\right)^2+\left(Q\sin\theta^2\right)\\\Rightarrow R^2=P^2+2PQ\cos\theta+Q^2\cos^2\theta+Q^2\sin^2\theta\\\Rightarrow R^2=P^2+2PQ\cos\theta+Q^2\left(\cos^2\theta+\sin^2\theta\right)\\\Rightarrow R^2=P^2+Q^2+2PQ\cos\theta\\\Rightarrow R=\sqrt{P^2+Q^2+2PQ\cos\theta}\end{array}\)

Thus, this is the magnitude of the resultant vector R.

Next we derive the direction for this resultant vector.

Let \( \phi \) be the angle made by resultant R with P. Then,

From \( \triangle OBC \)

\( \begin{array}{l}\ \ \ \tan\phi=\frac{BC}{OC}=\frac{BC}{OA+AC}\\\Rightarrow\tan\phi=\frac{Q\sin\theta}{P+Q\cos\theta}\\\Rightarrow\ \phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right]\end{array}\)

Thus, this is the magnitude of the resultant vector R.

Next we derive the direction for this resultant vector.

Let \( \phi \) be the angle made by resultant R with P. Then,

From \( \triangle OBC \)

\( \begin{array}{l}\ \ \ \tan\phi=\frac{BC}{OC}=\frac{BC}{OA+AC}\\\Rightarrow\tan\phi=\frac{Q\sin\theta}{P+Q\cos\theta}\\\Rightarrow\ \phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right]\end{array}\)

This is the direction of the resultant vector R.

Some Special Cases of Parallelogram Law of Vector Addition

We have read about the addition of two vectors according to the parallelogram law. Here we have considered that the direction of two vectors is the same and they have some angle θ between them.

The magnitude and direction of the resultant different possible cases of vector addition are tabulated below.

Vectors are Parallel (θ = 0°)

When two vectors are going in the same direction, we can add them directly. The final vector (called the resultant vector) will also go in the same direction and its size will be the total of both.

|R| = √(P² + Q² + 2PQcosθ)

Since, θ = 0°
⇒ |R| = √(P² + Q² + 2PQcos0)
⇒ |R| = √(P² + Q² + 2PQ)
⇒ |R| = √(P + Q)²
⇒ |R| = P + Q

So, when both vectors go in the same direction, we use R = P + Q where R is the final result, and P and Q are the two given vectors.

Vectors are in Opposite Direction (θ = 180°)

If the two vectors move in completely opposite directions, the final result (resultant vector) is the difference between them, because they work against each other.

|R| = √(P² + Q² + 2PQcosθ)

Since, θ = 180°
⇒ |R| = √(P² + Q² + 2PQcos180)
⇒ |R| = √(P² + Q² - 2PQ)
⇒ |R| = √(P - Q)²
⇒ |R| = P - Q

So, when they go in opposite directions, the final vector R is found using R = P - Q, where θ = 180°.

Vectors are Perpendicular (θ = 90°)

If two vectors are at right angles (90°) to each other, we use a special method based on the Pythagorean theorem. The final result (resultant vector) is the square root of the sum of the squares of the two vectors.

|R| = √(P² + Q² + 2PQcosθ)

Since, θ = 90°
⇒ |R| = √(P² + Q² + 2PQcos90)
⇒ |R| = √(P² + Q² + 0)
⇒ |R| = √(P² + Q²)

So, when two vectors are at 90°, the formula becomes R = √(P² + Q²), which gives us the size of the final vector.

Application of Parallelogram Law of Vector Addition

The Parallelogram Law of Vector Addition is used in many real-life and scientific situations. Here are some common ways it is applied:

• In Physics: This rule is used to find the total (resultant) force when two forces are acting at an angle to each other.
• In Navigation and Aviation: It helps figure out the final direction or speed of a boat, plane, or object when it moves both sideways and forward at the same time.
• In Flight Planning: Pilots use it to plan the best route by considering wind direction and plane speed.
• To Find Resultant Motion: When multiple forces act on an object together, this law helps us find the final direction and movement of the object.
• In Education: This law makes it easier for students to solve vector problems in mathematics and physics.
• In Astronomy: Scientists use this rule to study the movement of planets and stars. It helps them understand how gravity from different directions affects their path or orbit.

Steps to Use the Parallelogram Law of Vector Addition:

The parallelogram law helps us find the total effect (resultant) when two forces or vectors act at the same time from the same point. Follow these easy steps:

1. Draw the First Vector:
   Start by drawing the first vector using a ruler. Make sure you follow the correct direction and choose a suitable scale (for example, 1 cm = 2 N).
2. Draw the Second Vector:
   From the same starting point, draw the second vector at the correct angle and scale.
3. Form a Parallelogram:
   Now, using the two vectors as adjacent sides, complete a parallelogram by drawing lines parallel to each vector from the head of the other.
4. Draw the Diagonal:
   From the starting point where both vectors begin, draw a diagonal line across the parallelogram. This diagonal represents the resultant vector.
5. Understand the Result:
   The diagonal gives the magnitude (length) and direction of the total or combined effect of both vectors.

Parallelogram Law of Forces

We have seen that the parallelogram law helps us to find the resultant of vector addition. In physics, this process is also helpful for finding the resultant of two forces. Similar to that of vector addition, we add the forces here according to the formula.

Parallelogram Law of Forces states that if two concurrent forces, acting simultaneously on a body can be represented in magnitude and direction by the two sides of a parallelogram then their resultant is represented in magnitude and direction by the diagonal of the parallelogram, drawn from the same point of origin.

Here also the formula remains same, i.e.,

Magnitude of the resultant vector, \( \begin{array}{l}\left|R\right|=\sqrt{\left(P^2+Q^2+2PQ\cos\theta\right)}\end{array} \)

Direction of the resultant vector, \( \phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right] \)

Where, P and Q are the two concurrent forces and R is the resultant between them with angular direction \( \beta \)

For example, if two forces 5 N and 20 N are acting at an angle of \( 120° \) between them, then for calculating the resultant we do the following.

P=5N, Q=20N, \( 120° \)

\( \therefore\left|R\right|=\sqrt{P^2+Q^2+2PQ\cos\theta}=\sqrt{\left(5\right)^2+\left(20\right)^2+2\times5\times20\times\cos120}=\sqrt{25+400-100}=\sqrt{325}=18.03 \)

Thus the required resultant is 18.03N.

Next, \( \phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right]=\tan^{-1}\left[\frac{20\sin120}{5+20\cos120}\right]=\tan^{-1}\left(\frac{17.3205}{-5}\right)=\tan^{-1}\left(-3.4641\right)=73.8978° \)

Hence the required direction is \( 78.8978° \)

Triangle and Parallelogram Law of Vector Addition

The parallelogram law and the triangle law are two ways of combining vectors. Here are some ways they are different and similar:

How they work:

• Parallelogram Law: Imagine drawing a parallelogram using the vectors as sides. The resultant vector is like the diagonal of this shape.
• Triangle Law: Instead of a parallelogram, think of placing the vectors tip to tail. The resultant vector is the one that goes from the starting point of the first vector to the ending point of the last vector.

How they represent the answer:

• Parallelogram Law: The resultant vector is shown by the diagonal of the parallelogram. It passes through the point where the vectors intersect.
• Triangle Law: The resultant vector is represented by the vector that connects the starting point of the first vector to the ending point of the last vector.

Angles between the vectors:

• Parallelogram Law: The angles between the vectors used in the parallelogram law are determined by the angles between the sides of the parallelogram.
• Triangle Law: The angle between the vectors used in the triangle law is found by looking at the angle between the first and last vector.

Magnitude and direction:

Both the parallelogram law and the triangle law help find the magnitude and direction of the resultant vector.

When they are used:

• Parallelogram Law: This method works when adding two vectors.
• Triangle Law: The triangle law works when adding any number of vectors.

In short, the parallelogram law and the triangle law are two ways to add vectors visually. The parallelogram law involves making a parallelogram and using the diagonal, while the triangle law involves lining up the vector's tip to tail and drawing a vector from the starting point to the ending point. Both methods help find the size and direction of the resultant vector.

Important Notes of Parallelogram Law of Vector Addition

The important points that we need to keep in mind while applying the parallelogram law of vector addition.

• The two given vectors must be joined at the tails of each other and which then form the adjacent sides of a parallelogram.
• If the two given vectors are parallel in the same direction, then just add the magnitudes of the two vectors to get the required result.
• If the two given vectors are parallel, then the direction of the resultant can be found by the tan inverse of the division value of the two vectors.
• The triangle law and the parallelogram law of vector addition are equivalent to each other and they give the same value as the resultant vector. They are just two different methods of solving vector addition.
• The two given vectors and the resultant vector must all pass through the same point of origin of the parallelogram.

Parallelogram Law of Vector Addition Examples

Example 1: Two forces of magnitude 6N and 10N are inclined at an angle of 60° with each other. Calculate the magnitude of the resultant and the angle made by the resultant with 6N force. 

Solution: Let P and Q be two forces with magnitude 6N and 10N respectively and θ be an angle between them. Let R be the resultant force.

So, P = 6N, Q = 10N and \( \theta=60° \)

\( \begin{array}{l}\therefore\left|R\right|=\sqrt{P^2+Q^2+2PQ\cos\theta}\\\ \ \ \ \ \ \ \ \ \ =\sqrt{6^2+10^2+2\times6\times10\times\cos60}\\\ \ \ \ \ \ \ \ \ \ =\ \sqrt{36+100+120\times\frac{1}{2}}\\\ \ \ \ \ \ \ \ \ \ =\sqrt{196}\\\ \ \ \ \ \ \ \ \ \ =14\end{array}\)

Thus the required magnitude of their resultant vector is 14N.

Next for the direction.

Let ø be the angle between P and R. Then,

\( \begin{array}{l}\phi=\tan^{-1}\left[\frac{\left(Q\sin\theta\right)}{\left(P+Q\cos\theta\right)}\right]\\\ \ =\tan^{-1}\left[\frac{10\sin60}{6+10\cos60}\right]\\\ \ =\tan^{-1}\left[\frac{10\times\frac{\sqrt{3}}{2}}{6+10\times\frac{1}{2}}\right]\\\ \ =\tan^{-1}\left[\frac{5\sqrt{3}}{11}\right]\\\ \ =38.21321°\end{array}\)

This is the required angle for the direction of the resultant vector.

Example 2: Given two vectors 8î – 3ĵ + k̂ and 16î – 6ĵ + 2k̂ which are parallel, find the resultant vector for them.

Solution: Let, A = 8î – 3ĵ + k̂ and B = 16î – 6ĵ + 2k̂

We know that if two vectors are parallel, then the angle between them is zero and the resultant is just the sum of given vectors.

R = A + B
[∵ θ = 0]
So,
  = (8î – 3ĵ + k̂) + (16î – 6ĵ + 2k̂)
  = 24î – 9ĵ + 3k̂

R is the resultant addition for the two given vectors in the same direction.

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