Vectors

A vector is a magnitude together with a direction. The arrow you draw is just a picture of it, and the picture can start anywhere: an arrow from (0, 0) to (2, 5) and an arrow from (3, 1) to (5, 6) show the same vector, because both mean "go 2 right and 5 up." Vectors that agree in magnitude and direction are equal, wherever they are drawn.

That is the sharp line between a vector and a point. The point (2, 5) is a fixed address. The vector ⟨2, 5⟩ is an instruction, reusable from any starting address. Component form writes the instruction down: $\langle a, b \rangle$ means a units of horizontal change and b units of vertical change, signs carrying the directions.

The magnitude of ⟨a, b⟩ is the length of its arrow, and the arrow is the hypotenuse of a right triangle with legs |a| and |b|:

1. $|\langle a, b \rangle| = \sqrt{a^2 + b^2}$. For ⟨3, 4⟩: √(9 + 16) = 5, not 3 + 4 = 7.
2. Squares kill signs: ⟨−3, 4⟩ also has magnitude 5. Magnitude is never negative.
3. A unit vector has magnitude exactly 1. To point the same way as v with length 1, divide v by its magnitude: ⟨6, 8⟩ has magnitude 10, so its unit vector is ⟨0.6, 0.8⟩.

Adding the components is the single most common vector error, and it only agrees with the truth when one component is zero. Every diagonal vector is shorter than the add-the-legs guess, because the hypotenuse beats the detour.

Addition: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. Geometrically, chain the arrows tip to tail; the sum runs from the first tail to the last tip. Subtraction works the same way with a sign flip, and order matters: u − v points from v's tip toward u's tip, and v − u points the opposite way.

Scalar multiplication stretches without turning: kv keeps v's direction and scales its magnitude by |k| when k > 0. A negative scalar reverses the direction exactly, 180 degrees, no more and no less: −2⟨4, −1⟩ = ⟨−8, 2⟩, twice as long, pointing exactly opposite. The zero vector ⟨0, 0⟩ is what v + (−v) leaves behind: zero magnitude, no direction, and still a vector rather than the number 0.

One warning worth its own line: magnitudes do not add. Two vectors of magnitude 5 can sum to anything from magnitude 0 (opposite) to 10 (parallel). ⟨3, 4⟩ + ⟨−3, 4⟩ = ⟨0, 8⟩: two length-5 vectors with a length-8 sum. Only the components add; the lengths sort themselves out afterward.

The dot product multiplies matching components and adds the results: $\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$. The output is a scalar, a single number with no direction attached. ⟨2, 5⟩ · ⟨3, −1⟩ = 6 − 5 = 1. Not ⟨6, −5⟩; the products are summed, and the vector-ness is gone.

What the number says: positive when the vectors point broadly the same way, negative when they point broadly opposite ways, and exactly zero when they are perpendicular. ⟨1, 2⟩ · ⟨−4, 2⟩ = −4 + 4 = 0, so those two are perpendicular, no drawing needed. The dot product is the unit's one tool that converts geometry (angles) into arithmetic (a sign check).