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A 3D vector ⟨a, b, c⟩ has magnitude √(a² + b² + c²) and can be added, subtracted, and scalar-multiplied component-wise. Vectors represent displacement, force, velocity, and other directional quantities in 3D space.
When you learned 3D coordinate systems, you located points in space as ordered triples (x, y, z). Vectors in 3D extend this to describe *displacement* — how far and in which direction to move. Writing ⟨a, b, c⟩ means "move a units along x, b units along y, c units along z." The distinction between a point and a vector is subtle but important: a point is a location; a vector is a displacement that can originate anywhere.
The magnitude of a vector is its length — the actual distance of the displacement. Because the three axes are mutually perpendicular, you apply the Pythagorean theorem twice: first combine the x and y components into a horizontal distance √(a² + b²), then combine that with the z component to get the full 3D magnitude √(a² + b² + c²). This is a direct extension of the 2D distance formula, and the squaring step ensures the result is always non-negative regardless of the signs of the components.
Vector addition is computed component-wise because the axes are independent — what happens along x has no effect on y or z. If you walk 3 meters east and your friend walks 4 meters east, together you have covered 7 meters east; the east-direction displacements simply add. The same logic applies to each axis separately. Geometrically, this corresponds to the tip-to-tail rule: place the tail of the second vector at the tip of the first, and the vector from the original tail to the final tip is the sum.
Scalar multiplication stretches or shrinks a vector: k × ⟨a, b, c⟩ = ⟨ka, kb, kc⟩. For positive k, the direction is unchanged and the magnitude scales by k. For negative k, the direction reverses and the magnitude scales by |k|. These two operations — vector addition and scalar multiplication — are the building blocks for the entire space of linear algebra and multivariable calculus: dot products, cross products, projections, and gradient vectors all rest on this foundation.