Right Angle
GeometryA 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 $90^\circ$. It looks like the corner of a square. We mark right angles with a small square symbol in the corner instead of an arc.
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. The letter L also shows a right angle shape.
Key Insight
Right angles are everywhere in buildings and furniture because they create stable, balanced structures. Carpenters use a "square" tool specifically to check for right angles when building.
Definition
A right angle measures exactly $90^\circ$ ($\pi/2$ radians). Two lines forming a right angle are perpendicular. A right angle is the boundary between acute angles (less than $90^\circ$) and obtuse angles (greater than $90^\circ$). It is indicated by a small square at the vertex.
Example
In a coordinate plane, the x-axis and y-axis meet at a right angle. A right triangle has exactly one $90^\circ$ angle. The diagonal of a rectangle creates two right triangles. Adjacent sides of a rectangle are always perpendicular.
Key Insight
The Pythagorean theorem applies specifically to right triangles. The right angle is what makes the special relationship $a^2 + b^2 = c^2$ hold. Without a right angle, the law of cosines applies instead: $c^2 = a^2 + b^2 - 2ab\cos C$.
Definition
A right angle is an angle of $\pi/2$ radians. Two vectors $u$ and $v$ form a right angle iff $u \cdot v = 0$ (they are orthogonal). In a Euclidean space, orthogonality is the defining condition for perpendicularity and underpins the Gram-Schmidt process and orthogonal bases.
Example
Vectors $(1, 2)$ and $(-2, 1)$ are perpendicular: $(1)(-2) + (2)(1) = -2 + 2 = 0$. The projection of $u$ onto $v$ is zero when they are orthogonal. Orthogonal matrices preserve angles and lengths, making them essential in transformations.
Key Insight
Orthogonality in inner product spaces generalizes the right angle. Orthonormal bases (bases of mutually perpendicular unit vectors) simplify computations enormously. In quantum mechanics, orthogonal state vectors represent mutually exclusive measurement outcomes.