Vectors in the Plane and Space

Arrows with Coordinates

A vector is a quantity with both magnitude and direction, pictured as an arrow. In coordinates, a vector in the plane is an ordered pair $(v_1, v_2)$ and in space an ordered triple $(v_1, v_2, v_3)$; in general $\mathbb{R}^n$ has $n$-component vectors. The components are the signed displacements along each axis, and a vector is unchanged by sliding it (position does not matter — only length and direction).

Two operations are fundamental:

• Addition is componentwise: $(a_1, a_2) + (b_1, b_2) = (a_1 + b_1,\, a_2 + b_2)$. Geometrically, place the arrows tip-to-tail; the sum runs from the first tail to the last tip (the parallelogram rule).
• Scalar multiplication scales each component: $c(v_1, v_2) = (cv_1, cv_2)$. This stretches ($|c| > 1$) or shrinks ($|c| < 1$) the arrow, and a negative $c$ reverses its direction.

The magnitude (length, or norm) comes from the Pythagorean theorem: $\|v\| = \sqrt{v_1^2 + v_2^2}$ in the plane, and $\sqrt{v_1^2 + v_2^2 + v_3^2}$ in space. A unit vector has length 1 and encodes pure direction; dividing any nonzero vector by its length normalizes it to a unit vector.

These operations obey the same axioms across all dimensions: addition is commutative and associative, there is a zero vector, every vector has a negative, and scalars distribute. That uniformity is what lets the geometry of arrows in the plane carry over unchanged to spaces we cannot draw.

Common pitfall: adding vectors by adding their magnitudes, or confusing a vector's magnitude with its components. Vectors add componentwise — the length of a sum is generally not the sum of the lengths ($\|u + v\| \le \|u\| + \|v\|$, with equality only when they point the same way). And the magnitude is a single number $\sqrt{v_1^2 + v_2^2}$, not the list of components.