Vector and Linear Algebra Basics: Magnitude, Dot Product, and Lines
Vector magnitude, dot product, and slope-intercept form — three foundational calculations from linear algebra and coordinate geometry.
Vectors and lines are both described by coordinates — these three calculations extract the most commonly needed information from them.
Vector magnitude: the length of a vector
For a 3D vector, magnitude is the square root of the sum of squared components: √(x² + y² + z²). A vector (3, 4, 12) has a magnitude of √(9+16+144) = √169 = 13 — a direct extension of the Pythagorean theorem into three dimensions.
Dot product: measuring alignment between two vectors
The dot product multiplies corresponding components and sums them: (1,2,3) · (4,5,6) = 1×4 + 2×5 + 3×6 = 32. A dot product of zero means the vectors are perpendicular; the sign and magnitude of a nonzero result reveal how aligned two vectors are.
Slope-intercept form: describing a line from two points
Given two points, slope is (y₂−y₁)/(x₂−x₁) and the y-intercept follows from substituting one point back in. Points (1,4) and (3,8) give a slope of (8−4)/(3−1) = 2 and a y-intercept of 4 − 2×1 = 2, so the line is y = 2x + 2.
From coordinates to concrete answers
Whether the question is "how long is this vector," "how aligned are these two directions," or "what line passes through these points," these three calculations get straight to the number. Try the vector magnitude calculator, dot product calculator, and slope-intercept form calculator.