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Perpendicular vectors

Two vectors are perpendicular when their dot product is zero. That follows because the cosine of 90∘90^\circ is zero. If one vector has an unknown coordinate, you set the dot product equal to zero and solve. A normal vector of a line is perpendicular to the direction vector. You can check the answer by calculating the dot product again and seeing that it is zero.

Try it yourself

Vector addition

u⃗⊥v⃗⇔u⃗⋅v⃗=0\vec u \perp \vec v \Leftrightarrow \vec u \cdot \vec v = 0perpendicular
x1x2+y1y2=0x_1x_2 + y_1y_2 = 0the equation

Symbols

u⃗, v⃗\vec u,\ \vec vthe vectors

Example

[2,1]⋅[3,−6]=6−6=0[2, 1] \cdot [3, -6] = 6 - 6 = 0, so the vectors are perpendicular. For [2,t][2, t] and [3,−6][3, -6], 6−6t=06 - 6t = 0 gives t=1t = 1.

Zero in the dot product means 90∘90^\circ, not that the vectors are equal or parallel.
Practise vectors in the plane for free →

← The dot product · Ordered and unordered selections →

Part of Mathematics R1: Vectors in the plane.