# Two vectors calculator

For calculations implying one vector, see Vector calculator.

Use this online calculator to do operations on two vectors : addition, subtraction, scalar and cross products (defined for dimensions 3 and 7), formed angle by two vectors and projection of a vector on another vector.

## Dot product

If \vecu and \vecv are two vectors in the 3-dimensional space \mathbb{R^3} with the following coordinates :

\vecu = (x_1,x_2,x_3)

\vecv = (y_1,y_2,y_3)

then the dot product of \vecu by \vecv ca be written,

\vecu . \vecv = x_1.y_1 + x_2.y_2 + x_3.y_3

There is another definition using the vector norm and the angle \theta formed by vectors \vecu and \vecv: The dot product is then calculated as follows,

\vecu . \vecv = norm(u) . norm(v) . cos(\theta)

By the way, we can calculate the angle between the two vectors with the following formula,

\theta = arccos((\vecu . \vecv) / (norm(u) . norm(v)))

Example:

\vecu and \vecv are two vectors with the following coordinates :

\vecu = (1,4,-3)

\vecv = (10,2,2)

then the dot product of \vecu by \vecv is,

\vecu . \vecv = 1.10 + 4.2 + (-3).2 = 12

## Vector Projection

The vector projection of a vector \vecu on a non-zero vector \vecv is the orthogonal projection of \vecu to \vecv as shown in the diagram below (\vecu_1 being the projection of \vecu on \vecv). \vecu_1 is defined by:

proj_\vecv(\vecu) = \vecu_1 = \(vecu . \vecv)/norm(vecv)^2 . \vecv

Another formula:

The angle \theta formed by the vectors \vecu and \vecv can also be used. The projection of \vecu to \vecv can be defined as follows:

\vecu_1 = proj_\vecv(\vecu) = (norm(vecu).cos(\theta)) . \vecv / norm(v)