A Student Guide to Vectors

A vector is a quantity that has both magnitude and direction. Unlike scalars (like temperature or mass), vectors need both a size and a direction to be fully described. Velocity, force, and displacement are all vector quantities.

Vector Operations

Vector addition: add corresponding components. Scalar multiplication: multiply each component by the scalar. The dot product (also called scalar product) of two vectors gives a scalar: a · b = |a||b|cos(θ). It tells you how much two vectors point in the same direction. The cross product gives a vector perpendicular to both inputs.

Magnitude and Direction

The magnitude (length) of a vector [a, b, c] is √(a² + b² + c²). The unit vector in the same direction is v/|v|. The angle between two vectors can be found using the dot product formula.

Applications

Vectors are essential in physics (force diagrams, projectile motion), computer graphics (3D rendering, lighting calculations), machine learning (feature vectors), and engineering (structural analysis).

Use our Vector Calculator to compute dot products, cross products, and magnitudes.