Semester Exam Review - Chagrin Falls Schools
... ____ 33. Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements. If an elephant weighs more than 2,000 pounds, then it weighs more than Jill’s car. If something weighs more than Jill’s car, then it is too heavy for the bridge. Smiley the Elephant weig ...
... ____ 33. Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements. If an elephant weighs more than 2,000 pounds, then it weighs more than Jill’s car. If something weighs more than Jill’s car, then it is too heavy for the bridge. Smiley the Elephant weig ...
eUClidean geometrY
... endpoint A, is on the same side of �AB as C, and is perpendicular to � AB ; rC has endpoint C, is on the same side of � BC as A, and is perpendicular to � BC . then rA and rC intersect at a point D, and the angle fomed at this intersection, ∠ADC, is a right angle. therefore ABCD is a rectangl ...
... endpoint A, is on the same side of �AB as C, and is perpendicular to � AB ; rC has endpoint C, is on the same side of � BC as A, and is perpendicular to � BC . then rA and rC intersect at a point D, and the angle fomed at this intersection, ∠ADC, is a right angle. therefore ABCD is a rectangl ...
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.