Formulas & tables/Mathematics/Analytic geometry/Dot product

Dot product

u⃗∘v⃗=x1x2+y1y2=∣u⃗∣∣v⃗∣cos⁡α\vec{u}\circ\vec{v}=x_1x_2+y_1y_2=|\vec u||\vec v|\cos\alpha

Result unit: — (liczba)

What it describes

The dot product of two vectors; it equals zero exactly when the vectors are perpendicular.

Symbols

(x1,y1),(x2,y2)(x_1,y_1),(x_2,y_2)
coordinates of the vectors
∣u⃗∣,∣v⃗∣|\vec u|,|\vec v|
magnitudes of the vectors
α\alpha
angle between the vectors

Where it comes from

The dot product combines the magnitudes of the vectors and the angle between them. When it is zero, the vectors are perpendicular; a positive value means an acute angle, a negative value an obtuse one.

When to use it

Checking perpendicularity, computing the angle between vectors, projecting a vector.

Example

Given u⃗=[1,2]\vec u=[1,2], v⃗=[3,4]\vec v=[3,4]. Find the dot product.
Substitution1⋅3+2⋅41\cdot3+2\cdot4.
Resultu⃗∘v⃗=3+8=11\vec u\circ\vec v=3+8=11.

Related formulas

Magnitude of a vector Condition for perpendicular lines

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