Right Angle
A right angle measures exactly 90 degrees and is formed when two lines or rays are perpendicular to each other.
Formula
\text{angle} = 90^\circ = \pi/2 \text{ radians}
Definition
A right angle is exactly $90^\circ$ ($\pi/2$ radians), looking like the corner of a square; we mark it with a small square symbol instead of an arc. Two lines forming a right angle are perpendicular, and it is the boundary between acute angles (less than $90^\circ$) and obtuse angles (greater than $90^\circ$). Two vectors $u$ and $v$ form a right angle if and only if $u \cdot v = 0$ (they are orthogonal), a condition that underpins the Gram-Schmidt process and orthogonal bases in any Euclidean space.
Example
The corner of a piece of paper, the corner of a door, and the angle between a wall and the floor are all right angles. In a coordinate plane, the $x$-axis and $y$-axis meet at a right angle, and a right triangle has exactly one $90^\circ$ angle. Vectors $(1, 2)$ and $(-2, 1)$ are perpendicular: $(1)(-2) + (2)(1) = -2 + 2 = 0$.
Key Insight
Right angles are everywhere in buildings and furniture because they create stable, balanced structures, and carpenters use a "square" tool specifically to check for them. The Pythagorean theorem $a^2 + b^2 = c^2$ applies only because of the right angle; without one, the law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ takes over instead. Orthogonality in inner product spaces generalizes the right angle far beyond geometry: orthonormal bases simplify computation enormously, and in quantum mechanics orthogonal state vectors represent mutually exclusive measurement outcomes.