This **linear combination calculator** is used in **mathematics**, particularly in **linear algebra** and **vector spaces**, to help compute the result of combining multiple **vectors** or **scalars** through **addition** and **multiplication** operations.

Consider a scenario where you have two vectors:

v₁= (2, 3, 1)v₂= (4, -1, 5)

To find their **linear combination** with coefficients **a = 2** and **b = 3**, you would calculate:

**2v₁ + 3v₂ = 2(2, 3, 1) + 3(4, -1, 5) = (4, 6, 2) + (12, -3, 15) = (16, 3, 17)**

## Linear Combination Calculator

Vector 1 | Vector 2 | Coefficient 1 | Coefficient 2 | Result |
---|---|---|---|---|

(1, 2, 3) | (4, 5, 6) | 2 | 3 | (14, 19, 24) |

(-1, 0, 2) | (3, -2, 1) | 0.5 | -1 | (-3.5, 2, 0) |

(2, -1, 4) | (0, 3, -2) | -2 | 1.5 | (-4, 5.5, -11) |

(5, 5, 5) | (-1, -1, -1) | 1 | 10 | (-5, -5, -5) |

(0.5, 1.5, 2.5) | (1, 2, 3) | 4 | -0.5 | (1.5, 5, 8.5) |

(3, 1, 2) | (2, 4, 6) | 1 | 2 | (7, 9, 14) |

(0, 0, 0) | (1, 1, 1) | 3 | 1 | (3, 3, 3) |

(1, 2, 1) | (2, 0, 3) | -1 | 2 | (0, 2, 5) |

(1, 1, 1) | (1, 1, 1) | 0.5 | 0.5 | (1, 1, 1) |

(4, 0, -2) | (1, 3, 5) | 2 | -1 | (6, -3, 8) |

(2, 3, 4) | (5, -1, 2) | 0 | 4 | (20, -4, 8) |

(1, -1, 0) | (0, 2, 3) | 3 | 1 | (3, 5, 3) |

(6, 7, 8) | (1, 1, 1) | -2 | 1 | (4, 5, 6) |

(3, 3, 3) | (2, 2, 2) | 1 | 1 | (5, 5, 5) |

(1, 2, 3) | (-1, 0, 1) | 3 | 2 | (0, 6, 9) |

(0, 1, 2) | (3, 4, 5) | 1 | -1 | (-2, -3, -3) |

## Linear Combination Formula

For a set of vectors **v₁, v₂, …, vₙ** and corresponding scalar coefficients **a₁, a₂, …, aₙ**, the linear combination is expressed as:

**a₁v₁ + a₂v₂ + ... + aₙvₙ**

This formula can be expanded to show individual components:

`(a₁x₁ + a₂x₂ + ... + aₙxₙ, a₁y₁ + a₂y₂ + ... + aₙyₙ, a₁z₁ + a₂z₂ + ... + aₙzₙ)`

Let’s combine three vectors:

v₁= (1, 2, 3)v₂= (4, 5, 6)v₃= (7, 8, 9)

With coefficients **a₁ = 2**, **a₂ = -1**, and **a₃ = 3**, the linear combination is:

`2v₁ - 1v₂ + 3v₃ = 2(1, 2, 3) - 1(4, 5, 6) + 3(7, 8, 9)`

= (2, 4, 6) + (-4, -5, -6) + (21, 24, 27)

= (19, 23, 27)

## How to Find Linear Combination?

**Identify the Vectors**: Start with the vectors you want to combine. Let’s say **v1 = (2, 3) and v2 = (1, 4).**

**Set Up the Equation**: If you want to find a linear combination that equals b = (5, 10), set up the equation: **c1 * v1 + c2 * v2 = b**

**Formulate the System of Equations**: This leads to a system of equations based on the components:

`c1 * 2 + c2 * 1 = 5 (1)`

c1 * 3 + c2 * 4 = 10 (2)

**Solve the System**: Use methods such as substitution or elimination to solve for c1 and c2.

For example, from equation (1): c2 = 5 – 2 * c1 Substituting c2 into equation (2): 3 * c1 + 4 * (5 – 2 * c1) = 10

3 * c1 + 20 – 8 * c1 = 10

-5 * c1 = -10

c1 = 2 Substituting back to find c2: c2 = 5 – 2 * (2) = 1 Thus, c1 = 2 and c2 = 1 are the coefficients for the linear combination.

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