Step by step worksheet, 2d vectors

Solves **vectors** in this form... \[a\left(a_i,\ a_j,\ a_k\right)\]

Angle between two vectors calculator computes the angle between two vectors in a 2D space.

Follow steps below to solve **the angle between two vectors in a 2D space.**

Angle between two vectors calculator computes the angle between two vectors in a 2D space. A 2D Vector is a vector geometry in 2-dimensions running from point A (tail) to point B (head). Each vector has a magnitude (or length) and direction and can be calculated by taking the square root of the sum of each components in space.

To illustrate the calculation of the angle between two vectors in a 2D space, let's consider the following example:

Let's say we have two vectors in a 2D space:

Vector A with magnitude 3 and direction 45 degrees counterclockwise from the positive x-axis.

Vector B with magnitude 2 and direction 30 degrees counterclockwise from the positive x-axis.

**Step 1: Represent the vectors graphically**

Draw a coordinate system with the x-axis and y-axis. Place the tail of Vector A at the origin (0,0) and draw the arrow pointing towards point B. Similarly, place the tail of Vector B at the origin and draw the arrow pointing towards point C.

**Step 2: Calculate the components of each vector**

To calculate the components of Vector A, we need to determine its x-component (Ax) and y-component (Ay). Since Vector A is 45 degrees counterclockwise from the positive x-axis, we can use trigonometry to find these components:

**Ax = magnitude of A cos(angle of A)= 3 cos(45 degrees)= 3 * sqrt(2) / 2= 3sqrt(2) / 2**

**Ay = magnitude of A sin(angle of A)**

= 3 * sqrt(2) / 2

= 3sqrt(2) / 2

Similarly, we can calculate the components of Vector B:

**Bx = magnitude of B cos(angle of B)= 2 cos(30 degrees)= 2 * sqrt(3) / 2= sqrt(3)**

**By = magnitude of B sin(angle of B)= 2 sin(30 degrees)= 2 * 1/2= 1**

**Step 3: Calculate the dot product of the vectors**

The dot product of two vectors is calculated by multiplying their corresponding components and summing them up:

Dot product (A · B) = Ax * Bx + Ay * By

**= (3sqrt(2) / 2) sqrt(3) + (3sqrt(2) / 2) 1= 3sqrt(6) / 2 + 3sqrt(2) / 2= (3sqrt(6) + 3sqrt(2)) / 2**

**Step 4: Calculate the magnitudes of the vectors**

The magnitude of a vector is the length or size of the vector and can be calculated using the Pythagorean theorem:

Magnitude of A = sqrt(Ax^2 + Ay^2)

= sqrt((3sqrt(2) / 2)^2 + (3sqrt(2) / 2)^2)

= sqrt(9/2 + 9/2)

= sqrt(9)

= 3

Magnitude of B = sqrt(Bx^2 + By^2)

= sqrt((sqrt(3))^2 + 1^2)

= sqrt(3 + 1)

= sqrt(4)

= 2

**Step 5: Calculate the angle between the two vectors**

The angle between two vectors can be found using the dot product and the magnitudes of the vectors:

Angle between A and B (θ) = cos^(-1)((A · B) / (magnitude of A * magnitude of B))

= cos^(-1)(((3sqrt(6) + 3sqrt(2)) / 2) / (3 * 2))

= cos^(-1)((3sqrt(6) + 3sqrt(2)) / 6)

= cos^(-1)(sqrt(6) + sqrt(2)) / 2

Therefore, the angle between Vector A and Vector B in this example is cos^(-1)((sqrt(6) + sqrt(2)) / 2).

**θ = arccos( (a · b) / (|a| * |b|) )**

Step 1: Find the dot product of vectors a and b.

**a · b = (3 2) + (4 8)**

** = 6 + 32**

** = 38**

Step 2: Find the magnitude (length) of vector a.

**|a| = √(3^2 + 4^2)**

** = √(9 + 16)**

** = √25**

** = 5**

Step 3: Find the magnitude (length) of vector b.

**|b| = √(2^2 + 8^2)**

** = √(4 + 64)**

** = √68**

** ≈ 8.246**

Step 4: Substitute the values in the formula to find the angle.

**θ = arccos( 38 / (5 * 8.246) )**

Step 5: Use a calculator to find the arccos of the number.

**θ ≈ arccos(0.91892)**

** ≈ 23.57 degrees (rounded to two decimal places)**

Therefore, the angle between vectors a(3, 4) and b(2, 8) in 2D space is approximately 23.57 degrees.

- AI Math Formula
- AI Graduate Homework Help
- AI Physics Mechanics Solver
- AI Math Solver
- Math
- Step By Step Math Solver
- AI Calculators, Over 100 AI Calculator
- Welcome
- AI Economics Solver Microeconomics
- AI Chemistry Solvers
- Register
- Confirm
- MathCrave Math Sheets
- formula1
- AI Math Solvers
- AI Physics Thermodynamics Solver
- AI Chemistry: General Chemistry
- AI Physics Waves and Optics Solver
- Logarithms Solvers
- Geometric Formulas, Common Shapes
- Income Calculator
- Quadratic Equation Calculator Worksheet
- Calculus Worksheets Generator
- Math Calculators
- Greek Math Symbols
- Mortgage Calculator
- Vector Cross Product Calculator
- Vector Addition Calculator
- Takt Time Calculator
- Sum of Infinite Geometric Series
- Sum of Harmonic Progression Series
- Sum of a GP
- Standard Deviation Calculator
- Quadratic Equation Problem Solver
- Regression Coefficients
- Profit and Loss Calculator
- Prime Number Calculator
- Permutation Solver
- Percentage Subtraction Calculator
- Percentage Quantity Calculator
- Percentage Multiplication Calculator
- Percentage of Calculator
- Percentage Increase Calculator
- Percentage Error Calculator
- Percentage Change Calculator
- Orthocenter of Triangle Calculator
- Multiplication Table Generator
- Mixed Number to Improper Fraction
- MidRange
- Matrix Subtraction Calculator
- Matrix Multiplication Calculator
- Matrix Addition Calculator
- Logarithms Rules
- Logarithms Product Rule
- Logarithms Number
- Logarithms Calculator With Steps
- Logarithms Base
- Logarithms - Indicial Equation Calculator
- Logarithm Power Rule
- Logarithm Rules, Quotient
- Log x Calculator
- Linear Equation Calculator
- Leap Year Calculator
- Inverse Proportion Calculator
- LCM HCF Prime Factorization Calculator
- Inverse 3 x 3 Matrix Solver
- Inverse of Matrix 2 by 2 Calculator
- Improper Fraction to Mixed Number
- Integration Solver
- GST Tax Calculator
- Graph Plotter Calculator
- Fractions Calculator
- Fraction Multiplication
- Fraction Division
- Fraction Addition, Subtraction
- Express as Percentage
- EMI Calculator
- Effective Interest Rate Calculator
- Dot Product Two Dimensional
- Dot Product Calculator
- Correlation Coefficients Calculator
- Currency Converter
- Determinant of a 3x3 Matrix Calculator
- Convert Spherical to Cylindrical
- Direct Proportion Calculator
- Convert Cartesian to Cylindrical
- Conversion Calculator
- Compound Interest Calculator
- Combination Solver
- Complex Numbers Solver
- Circumcenter of a Triangle
- Calculus Step by Step
- Calculus Calculator
- BMI Calculator
- Arithmetic Progression Terms Calculator
- Area of Polygon Solver
- Apothem of Pentagon
- Angle Plotter Calculator
- Angle of Elevation Solver
- Angle of Depression Solver
- Angle between Vectors in 3D
- Angle Between Two Vectors in a 2D
- 2 by 2 Matrix Determinants Calculator
- AI Hyperbolic Functions Solver
- AI Binomial Series
- AI Partial Fractions Solver
- AI Inequalities Solver
- AI Polynomial Division Solver
- About Mathcrave
- Privacy Policy
- AI Simultaneous Equations Solver
- AI Physics Quantum Mechanics Solver
- AI Physics Electricity and Magnetism Solver
- AI Transpose Formulae Solver
- AI Linear Equation Solver
- AI Powers and Roots Solver
- MathCrave AI Fractions
- Contact Us
- Calculators
- AI Ratio and Proportions Solver
- AI Decimals Solver
- AI Fraction Solver
- AI Basic Arithmetic Solver
- Volume of a Cylinder Calculator

Mathcrave is an online math solver offering a wide range of free math worksheets on calculus, algebra, physics and more for free,

©2024 MathCrave- All rights reserved