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Transformations: translation, rotation, reflection, dilation

Rigid motions and scalings, and the matrices that perform them.

Translations slide, rotations turn, reflections flip, dilations scale. The first three preserve distances (isometries); dilation preserves shape. Picture it: the 2×2 matrix [[0,−1],[1,0]] turns every vector a quarter turn. The 3D view shows the unit square rotating. Think it: transformations compose, invert and form a group, which is where abstract algebra begins.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · inverse of [[0,-1],[1,0]]

Linear-Algebra · Matrix-Inverse

Inverse of [[0,-1],[1,0]]

\left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]

ជំហាន​ដោយ​ជំហាន

  1. \det A = 1

    A matrix is invertible only when its determinant is non-zero.

  2. A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{1}\left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

    For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.

  3. A^{-1} = \left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

បង្ហាញ​ចម្លើយ
A^{-1} = \left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

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ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: Transformations: translation, rotation, reflection, dilation

  1. Identify the transformation type from what is preserved: lengths (isometry) or only angles (similarity).
  2. Write it as a matrix acting on column vectors, plus a translation vector if there is one.
  3. Apply it to each vertex; compose by multiplying matrices in the order the moves happen (right to left).

សំណួរ​ដែល​មនុស្ស​សួរ

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

បន្ថែម​ទៀត​ក្នុង Geometry

TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryEuclid's axioms and the structure of proofCongruent and similar trianglesCircle theoremsSolid geometry: prisms, pyramids, spheres