Angle Between Two Vectors Formula

Formula for Finding the Angle Between Two Vectors:

The formula to calculate the angle (θ) between two vectors A and B is given by:

COS(θ) =A.B/ |A|.|B|

Where:

• A  and  B  are the two vectors.
• A * B  represents the dot product of vectors A and B.
• |A|  and  |B|  are the magnitudes (or lengths) of vectors A and B, respectively.

Explanation of the Formula:

The dot product  A * B  quantifies how much the two vectors are aligned with each other. When  A * B  is positive, the vectors are pointing in a similar direction, resulting in an acute angle.

The magnitudes  |A|  and  |B|  represent the lengths of the vectors. These values are always positive.

By dividing the dot product by the product of magnitudes, we normalize the result and obtain the cosine of the angle  θ .

Taking the inverse cosine (arc cosine) of this value gives us the actual angle  θ .

Example:

Let's consider an example to illustrate the formula. Suppose we have two vectors:

A = [3, 4]  and  B = [1, 2]

First, we find the dot product  A * B :

A * B = (3 * 1) + (4 * 2) = 3 + 8 = 11 *

Next, calculate the magnitudes of vectors A and B:

|A| = √((3^2 + 4^2)) = \√(9 + 16) = \√(25) = 5

|B| = √((1^2 + 2^2)) = \√(1 + 4) = \√(5)

Now, we can use the formula to find the angle  θ :

COS(θ) =A.B/ |A|.|B| =11/5 _* √5

Taking the inverse cosine:

θ = COS ^-1 ( 11/5 _* √5 )

Calculate  θ  using a calculator to find the angle between the two vectors

Also Read - Cylindrical Capacitor Formula

Some Solved Problems

Problem 1:

Given vectors A = [3, 4] and B = [1, 2], find the angle (θ) between them.

cos(θ) = (A · B) / (|A| · |B|)

cos(θ) = ((3 * 1) + (4 * 2)) / (√(3² + 4²) · √(1² + 2²))

cos(θ) = (3 + 8) / (√(9 + 16) · √(1 + 4))

cos(θ) = 11 / (5 * √5)

θ = cos⁻¹(11 / (5 * √5))

Problem 2:

Find the angle between the force vectors F₁ = [6, -2] and F₂ = [-3, 4].

cos(θ) = (F₁ · F₂) / (|F₁| · |F₂|)

cos(θ) = ((6 * -3) + (-2 * 4)) / (√(6² + (-2)²) · √((-3)² + 4²))

cos(θ) = (-18 - 8) / (√(36 + 4) · √(9 + 16))

cos(θ) = -26 / (5 * √5)

θ = cos⁻¹(-26 / (5 * √5))

Problem 3:

In a 2D space, find the angle (θ) between vectors (2, 3) and (-1, 5).

cos(θ) = (2 * -1 + 3 * 5) / (√(2² + 3²) * √((-1)² + 5²))

cos(θ) = (-2 + 15) / (√(4 + 9) * √(1 + 25))

cos(θ) = 13 / (√13 * √26)

θ = cos⁻¹(13 / (√13 * √26))

Problem 4:

Calculate the angle (θ) between vectors (1, 0, 2) and (-2, 1, 3) in 3D space.

cos(θ) = (1 * -2 + 0 * 1 + 2 * 3) / (√(1² + 0² + 2²) * √((-2)² + 1² + 3²))

cos(θ) = (-2 + 6) / (√(1 + 0 + 4) * √(4 + 1 + 9))

cos(θ) = 4 / (2 * √14)

θ = cos⁻¹(4 / (2 * √14))

Problem 5:

Find the angle (θ) between velocity vectors V₁ = [5, 2] and V₂ = [-3, -1].

cos(θ) = (V₁ · V₂) / (|V₁| * |V₂|)

cos(θ) = ((5 * -3) + (2 * -1)) / (√(5² + 2²) * √((-3)² + (-1)²))

cos(θ) = (-15 - 2) / (√(25 + 4) * √(9 + 1))

cos(θ) = -17 / (√29 * √10)

θ = cos⁻¹(-17 / (√29 * √10))

Problem 6:

Calculate the angle (θ) between geographic coordinates (40° N, 75° W) and (30° N, 80° W).

cos(θ) = sin(40°) * sin(30°) + cos(40°) * cos(30°) * cos(75° - 80°)

cos(θ) = (0.6428 * 0.5) + (0.7660 * 0.8660 * 0.0872)

cos(θ) = 0.3214 + 0.0582

θ = cos⁻¹(0.3214 + 0.0582)

Problem 7:

Given vectors C = [2, 1] and D = [-1, -2], determine the angle (θ) between them.

cos(θ) = (C · D) / (|C| * |D|)

cos(θ) = ((2 * -1) + (1 * -2)) / (√(2² + 1²) * √((-1)² + (-2)²))

cos(θ) = (-2 - 2) / (√(4 + 1) * √(1 + 4))

cos(θ) = -4 / (√5 * √5)

θ = cos⁻¹(-4 / (√5 * √5))

Problem 8:

Find the angle (θ) between the vectors P = [3, 2] and Q = [-1, 4].

cos(θ) = (P · Q) / (|P| * |Q|)

cos(θ) = ((3 * -1) + (2 * 4)) / (√(3² + 2²) * √((-1)² + 4²))

cos(θ) = (-3 + 8) / (√(9 + 4) * √(1 + 16))

cos(θ) = 5 / (√13 * √17)

θ = cos⁻¹(5 / (√13 * √17))

Problem 9:

Calculate the angle (θ) between vectors M = [-2, 3, -1] and N = [1, -2, 2] in 3D space.

cos(θ) = (M · N) / (|M| * |N|)

cos(θ) = ((-2 * 1) + (3 * -2) + (-1 * 2)) / (√((-2)² + 3² + (-1)²) * √(1² + (-2)² + 2²))

cos(θ) = (-2 - 6 - 2) / (√(4 + 9 + 1) * √(1 + 4 + 4))

cos(θ) = -10 / (√14 * √9)

θ = cos⁻¹(-10 / (√14 * √9))

Also Read - Position Formula

Some Important Formulas

1. Angle Between Two Forces:

To find the angle between two forces, use the formula:

cos(θ) = (F₁ · F₂) / (|F₁| · |F₂|)

where θ is the angle, F₁ and F₂ are the force vectors, and |F₁| and |F₂| are their magnitudes.

2. Angle in 2D Space:

In a 2D plane, the angle between two vectors (x₁, y₁) and (x₂, y₂) is given by:

cos(θ) = (x₁x₂ + y₁y₂) / (√(x₁² + y₁²) · √(x₂² + y₂²))

3. Angle in 3D Space:

In 3D space, to find the angle between vectors (x₁, y₁, z₁) and (x₂, y₂, z₂), use:

cos(θ) = (x₁x₂ + y₁y₂ + z₁z₂) / (√(x₁² + y₁² + z₁²) · √(x₂² + y₂² + z₂²))

4. Angle Between Velocity Vectors:

When dealing with velocity vectors in physics, the angle between two velocities (v₁, v₂) can be calculated using:

cos(θ) = (v₁ · v₂) / (|v₁| · |v₂|)

5. Angle in Geographic Coordinates:

In geography, the angle between two geographic coordinates (lat₁, lon₁) and (lat₂, lon₂) on Earth's surface can be determined by:

cos(θ) = sin(lat₁) · sin(lat₂) + cos(lat₁) · cos(lat₂) · cos(lon₂ - lon₁)

Also Read - Escape Speed Formula

What Are Vectors?

Definition of Vectors:

A vector is a mathematical entity that represents a physical quantity with both magnitude and direction. It is often denoted by an arrow pointing in a specific direction and has a length (magnitude) that represents the quantity's size or intensity.

Key Characteristics of Vectors:

1. Magnitude: Vectors have a magnitude, which is a nonnegative scalar quantity that indicates the size or intensity of the quantity being represented. The magnitude is usually denoted by the absolute value of the vector.
2. Direction: Vectors also have direction, which indicates the way in which the quantity is oriented or pointed. Direction can be described using angles, compass directions, or other coordinate systems.
3. Representation: Vectors are typically represented by arrows. The length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector.
4. Mathematical Notation: Vectors are often denoted by boldface letters (e.g., A), lowercase letters with an arrow on top (e.g., 𝑎→), or as ordered pairs (e.g., (x, y, z) for 3D vectors). The notation may vary depending on the context.

Examples of Vectors:

1. Displacement Vector : In physics, a displacement vector represents the change in position of an object. It has both magnitude (the distance traveled) and direction (the path taken).
2. Velocity Vector: Velocity is a vector that represents an object's speed and direction of motion. It is used in kinematics to describe how an object is moving.
3. Force Vector: In mechanics, force vectors represent the strength and direction of a force applied to an object. They are essential for understanding how objects move under the influence of forces.
4. Electric Field Vector : In electromagnetism, electric field vectors describe the direction and strength of electric fields at different points in space.
5. Vector in Computer Graphics: Vectors are used extensively in computer graphics to represent points, directions, and transformations. They are used to draw lines, curves, and shapes on screens.

Operations with Vectors:

Vectors can undergo various mathematical operations, including:

Vector Addition : Combining two vectors to obtain a resultant vector. This can be done geometrically using the parallelogram law or algebraically by adding corresponding components.

Scalar Multiplication: Multiplying a vector by a scalar (a single real number) changes its magnitude without altering its direction.

Dot Product: Calculating the dot product of two vectors results in a scalar quantity. It is used to find the angle between two vectors and determine how much they align with each other.

Cross Product: Calculating the cross product of two vectors results in a new vector that is perpendicular to the plane formed by the original vectors. It is commonly used in physics and engineering.

Applications of Vectors:

Vectors are applied in numerous fields, including:

Physics: Vectors are used to describe physical quantities such as displacement, velocity, acceleration, force, and electric fields.

Engineering: Engineers use vectors to analyze structural forces, design electrical circuits, and simulate fluid flow in various systems.

Computer Science: Vectors are used in computer graphics, data analysis, machine learning, and game development.

Mathematics: Vectors are a fundamental concept in linear algebra, providing a basis for understanding vector spaces, linear transformations, and eigenvalues.

Navigation: Vectors are essential in navigation systems, such as GPS, for determining positions and directions.

Vectors are mathematical objects that play a crucial role in representing physical quantities with both magnitude and direction. They are fundamental in various scientific, engineering, and computational disciplines, enabling us to model and understand a wide range of phenomena in the physical world.