Vectors in Three Dimensions
Vectors are arrows with length and direction, and two products called dot and cross turn geometric questions into simple arithmetic.
Components and magnitude
A vector in space is a triple (a, b, c) giving its reach along each axis, and its length is the square root of a squared plus b squared plus c squared. For example, the vector (1, 2, 2) has length equal to 3 by the Pythagorean theorem in three dimensions. Dividing a vector by its own length gives a unit vector pointing the same way, here (1/3, 2/3, 2/3).
\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}The dot product
The dot product multiplies matching components and adds them, producing a single number. Geometrically it equals the product of both lengths times the cosine of the angle between them, so it measures alignment. A zero dot product means the vectors are perpendicular, for example (1, 0, 0) dotted with (0, 1, 0) gives 0.
\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}|\,|\mathbf{v}|\cos\thetaThe cross product
The cross product of two vectors is a new vector perpendicular to both, computed from a determinant of the components. Its length equals the area of the parallelogram the two vectors span, so it also measures how far out of line the inputs are. The basic pattern is i cross j equals k, cycling through the three axes.
\mathbf{u} \times \mathbf{v}Flashcard
Length of a vector from its components
Check your understanding
Compute the dot product of the vectors (2, 1, 0) and (-1, 3, 4).