The Cauchy-Schwarz Inequality for Vectors in the Plane

●

If

u

and

v

are vectors in Euclidean space, then a special case of the Cauchy–Schwarz inequality states that



u

·

v

≤

u

||

v



. Drag the gray dots to change the positions of the two 2D vectors and compare the two sides of this inequality. The red dashed line is drawn at the angle between the two vectors (marked as a red arc on the left-hand image), which determines the relationship between the two sides of the inequality.