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Length and dot product in space

In space a vector has three coordinates. The length is the square root of the sum of the squares, an extension of Pythagoras. The dot product is the sum of the products of the matching coordinates, just as in the plane. It is zero when the vectors are perpendicular. You use the length for distances and the dot product for angles between vectors in space.

∣[x,y,z]∣=x2+y2+z2|[x,y,z]| = \sqrt{x^2+y^2+z^2}length
u⃗⋅v⃗=x1x2+y1y2+z1z2\vec u\cdot\vec v = x_1x_2+y_1y_2+z_1z_2dot product

Symbols

x, y, zx,\ y,\ zthe coordinates

Example

The length of [2,3,6][2, 3, 6] is 4+9+36=7\sqrt{4 + 9 + 36} = 7. [1,2,2]⋅[4,−1,3]=4−2+6=8[1, 2, 2] \cdot [4, -1, 3] = 4 - 2 + 6 = 8.

Remember the third coordinate. Without it you are calculating as if the vector lay in the plane.
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Part of Mathematics R2: Vectors in space.