Recipe

Test two vectors for orthogonality

Use when: you're given two vectors — possibly with an unknown entry — and asked whether they're orthogonal, or what value would make them orthogonal

Goal: a definite yes-or-no on orthogonality, or the exact unknown value that forces it, backed by a dot product that actually equals zero

Taught in Orthogonality

1. 1

   Pair the vectors entry by entry — first with first, second with second, and so on — and multiply each pair. The same rule applies no matter how many entries the vectors have.

   Example

   For \mathbf u=(3,4) and \mathbf v=(8,-6): the matching pairs are (3)(8) and (4)(-6).

   Why: the dot product is defined entry by entry — nothing later in this recipe means anything until each matching pair has actually been multiplied.

2. 2

   Sum every product from Step 1 into a single result.

   Example

   (3)(8) + (4)(-6) = 24 + (-24)
   • every entry of both vectors is a known number — the sum comes out as a plain number; move to Step 3's first branch.
   • one vector has an unknown entry (like k) — the sum comes out as an algebraic expression in that unknown; move to Step 3's other branch.

   Why: this sum IS the dot product \mathbf u\cdot\mathbf v — the single quantity the whole orthogonality test hinges on.

3. 3

   If the sum from Step 2 is a plain number, the vectors are orthogonal exactly when that number is 0. If it's an expression in an unknown, set the expression equal to 0 and solve for the unknown.

   Example

   24+(-24)=0, so \mathbf u\cdot\mathbf v=0 and \mathbf u\perp\mathbf v.

   Why: orthogonality is DEFINED as \mathbf u\cdot\mathbf v=0 — there's no other condition to check, and no partial credit for a dot product that's merely small.

4. 4

   Recompute the dot product with the numbers you now have (including any solved unknown) and confirm it comes out to exactly 0.

   Example

   (3)(8)+(4)(-6) = 24-24=0 ✓, confirming the conclusion from Step 3.

   Why: this catches an arithmetic or sign slip from Steps 1–3 before the orthogonality conclusion gets used downstream, such as building an orthogonal basis or a projection.