Vectors

Vector length

$$l = \sqrt {x^{2}+y^{2}}$$

x, y - vector coordinates

Find It is known that:

Calculate 'l'

Spatial vector length

$$l = \sqrt {x^{2}+y^{2}+z^{2}}$$

x, y, z - vector coordinates

Find It is known that:

Calculate 'l'

Scalar (dot) product of vectors

$$A\cdot B = a\cdot b\cdot cos(\alpha)$$

a, b - vector lengths
α - the angle between the vectors

Find It is known that:

Calculate 'A'

Scalar (dot) product of vectors according to the coordinates

$$A\cdot B = x_1\cdot x_2+y_1\cdot y_2$$

x1, y1 - first vector coordinates
x2, y2 - second vector coordinates

Find It is known that:

Calculate 'A'

Scalar (dot) product of spacial vectors according to the coordinates

$$A\cdot B = x_1\cdot x_2+y_1\cdot y_2+z1\cdot z2$$

x1, y1, z1 - first vector coordinates
x2, y2, z2 - second vector coordinates

Find It is known that:

Calculate 'A'

Scalar (dot) product of vertical vectors

$$x_1\cdot x_2+y_1\cdot y_2 = 0$$

x1, y1 - first vector coordinates
x2, y2 - second vector coordinates

Find It is known that:

Calculate 'x1'

Scalar (dot) product of spatial vertical vectors

$$x_1\cdot x_2+y_1\cdot y_2+z1\cdot z2 = 0$$

x1, y1, z1 - first vector coordinates
x2, y2, z2 - second vector coordinates

Find It is known that:

Calculate 'x1'

The angle between the vectors

$$cos(\alpha) = \frac{x_1\cdot x_2+y_1\cdot y_2}{\sqrt {x_1^{2}+y_1^{2}}\cdot \sqrt {x_2^{2}+y_2^{2}}}$$

α - angle between the vectors
x1, y1 - first vector coordinates
x2, y2 - second vector coordinates

Find It is known that:

Calculate 'α'

The angle between the spacial vectors

$$cos(\alpha) = \frac{x_1\cdot x_2+y_1\cdot y_2+z1\cdot z2}{\sqrt {x_1^{2}+y_1^{2}+z1^{2}}\cdot \sqrt {x_2^{2}+y_2^{2}+z2^{2}}}$$

α - angle between the vectors
x1, y1, z1 - first vector coordinates
x2, y2, z2 - second vector coordinates

Find It is known that:

Calculate 'α'

Collinear vectors

$$\frac{x_1}{x_2} = \frac{y_1}{y_2}$$

x1, y1 - first vector coordinates
x2, y2 - second vector coordinates

Find It is known that:

Calculate 'x1'

The distance between points

$$AB = \sqrt {(x_2-x_1)^{2}+(y_2-y_1)^{2}}$$

x1, y1 - first point coordinates
x2, y2 - second point coordinates

Find It is known that:

Calculate 'AB'

The distance between points in space

$$AB = \sqrt {(x_2-x_1)^{2}+(y_2-y_1)^{2}+(z2-z1)^{2}}$$

x1, y1, z1 - first point coordinates
x2, y2, z2 - second point coordinates

Find It is known that:

Calculate 'AB'