$Vector Operations$

Vector Operations

Add and subtract vectors, multiply a vector by a number and find a linear combination of vectors from coordinates with a step-by-step solution.

What is known:

Sum of vectors a + b

Difference of vectors a − b

Scalar multiplication k · a

Linear combination α · a + β · b

Space: in the plane (x, y) in space (x, y, z)

Coordinates are whole numbers, decimals or fractions: 3, −1.5, 2/3.

Test yourself

The vector operations calculator adds and subtracts vectors, multiplies a vector by a number and finds the linear combination αa + βb in the plane and in space. The length of the resulting vector is calculated right away. Coordinates can be whole numbers, decimals and fractions.

Operations in coordinates

Operation	Formula
Sum	a + b = (x₁ + x₂, y₁ + y₂, z₁ + z₂)
Difference	a − b = (x₁ − x₂, y₁ − y₂, z₁ − z₂)
Scalar multiplication	ka = (kx, ky, kz)
Linear combination	αa + βb = (αx₁ + βx₂, αy₁ + βy₂, αz₁ + βz₂)

Example. a = (1, 2, 3), b = (4, −1, 2).
a + b = (5, 1, 5), a − b = (−3, 3, 1)
2a − 3b = (2, 4, 6) + (−12, 3, −6) = (−10, 7, 0)

Geometric meaning

The sum follows the triangle rule (the second vector starts at the end of the first) or the parallelogram rule (the diagonal of the parallelogram on the vectors).
The difference a − b is the vector from the end of b to the end of a when both start at one point.
Multiplying by a number stretches the vector |k| times; for k < 0 the direction is reversed.