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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.

Umzekelo osebenzelayo · 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]

Inyathelo ngenyathelo

  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]

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

ngoku Khetha ingxaki, okanye ubhale okanye uzobe yakho. Inyathelo ngalinye, umfanekiso, impendulo ifihlakele de ubuze.

Ukugcina umsebenzi wakho

I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.

Bhalisa Igama elithile

Iimpawu ezisetyenziswa apha

Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.

Indlela yokusebenza: 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).

Imibuzo abantu bebuza

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.

IiNkqubo Geometry

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