Vectors and components: keeping direction in the calculation

A geometric vector describes a directed displacement. Drawing its tail at another location does not change the displacement if its length and direction remain the same. To calculate, choose perpendicular unit axes and list how much displacement lies along each one. Those numbers are components in the chosen basis. Rotating the axes changes the component list while leaving the underlying geometric displacement unchanged.

A vector's magnitude is its length, a nonnegative scalar. In an orthonormal Cartesian basis, the Pythagorean theorem computes length from the components. That assumption about the basis matters: if basis vectors are not perpendicular unit vectors, blindly squaring and summing coordinates generally gives the wrong geometric length. We use orthonormal coordinates throughout this introductory lesson so that ordinary Euclidean geometry applies.

To add two displacements, place the second arrow's tail at the first arrow's head. The resulting arrow runs from original start to final end. In a common basis, this operation adds matching components. It is commutative because the same two moves in reversed order reach the same endpoint in flat space. The path and total traveled distance can differ from the magnitude of the resultant.

Multiplying a vector by a positive scalar stretches it without changing direction. A negative scalar also reverses direction, and zero produces the zero vector. Unit vectors separate direction from size. Divide a nonzero vector by its magnitude to make a unit vector; the zero vector has no defined direction and cannot be normalized this way. Do not let a software division hide this geometric restriction.

The dot product multiplies corresponding components and adds the products. Geometrically it equals the product of the lengths times the cosine of their included angle. It is positive for acute alignment, zero for perpendicular nonzero vectors, and negative for obtuse alignment. Unlike a vector sum, its result is a scalar. Its physical unit is the product of the units carried by its inputs.

Dotting a vector with a unit direction gives its signed component along that direction. This explains why mechanical work uses force dotted with displacement: a perpendicular force contributes no work along that displacement. In data analysis a normalized dot product can measure similarity, but that interpretation depends on the chosen representation and metric. A numerical similarity is not automatically a statement about semantic meaning or causation.