The Zero-Vector Trap — Vectors (Mathematics)

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Watch Out

The Dot-Product = 0 Trap

$\vec{a} \cdot \vec{b} = 0$ means PERPENDICULAR, not that one of the vectors is zero. Two non-zero vectors absolutely can have dot product zero — that's what perpendicular means.

a \cdot b = 0 ⟺ a ⊥ b (both non-zero)

Zero dot product ≠ zero vectors

Trap

'Vectors must be zero'

Wrong! Two non-zero vectors can have $a \cdot b = 0$

Truth

Dot product = 0

Means the angle between them is $90°$

Test

$\overset{}{^} = (1, 0), \overset{}{^} = (0, 1)$

Both non-zero, but $\overset{}{^} \cdot \overset{}{^} = 0$