(a) The new coordinates geometrically if X represents the point (0, 2) m and this point is rotated about the origin 45∘ clockwise and then translated 2 units to the right and 3 units upward. (b) The value of Y=ABX, and explain the result. (c) If ABX equal to BAX. Interpret the resul. (d) A matrix that translate Y back to X

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(a) The new coordinates geometrically if X represents the point (0, 2) m and this   point is rotated about the origin 45∘ clockwise and then translated 2 units to the right and 3 units upward.   (b) The value of Y=ABX, and explain the result.  (c) If ABX equal to BAX. Interpret the resul.  (d) A matrix that translate Y back to X

d2saint0 · Accepted Answer

Given:  The matrices  A=[102013001], B=[12120−12120001]  Concept used:  When a point is rotated an angle theta clockwise about the origin, the transforming matrix is   [cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001]  To translate a point (x, y) horizontallyh units and vertically k units, we use the transformation matrix  [10h01k001]  Calculation:  (a) Let the point (0, 2) be represented by column matrix  X=[121]  When it is rotated 45∘ about the origin, then we have  [12120−12120001][021]=[111]  So after rotating 45∘ about the origin, the new coordinates are (1, 1).  Now we need to translate this point 2 units to the right and 3 units upward, so using above concept we have  [102013001][111]=[341]  Hence after rotating 45∘ about origin and then translating 2 units to the right and 3 units upward, we reach to the point (3, 4).  (b) Now to compure Y=ABX, we have  Y=ABX=[102013001][12120−12120101][021]  =[102013001][111]  =[341]  It is same as the above transformations.  Hence the first matrix A represents the translation of 2 units the right and 3 units upward.  And the matrix B represent the rotation of 45∘ clockwise about the origin of the point X.  (c) Now to check if ABX equal to BAX, we compute BAX first. So we have  BAX=[11120−12120001][23 + 21][5 + 221 + 221]  We can see that ABX is not equal to BAX. It implies that if we translate the point X, 2 units to the right and 3 units upward first and then rotate about the origin 45∘ clockwise we will not reach to the same point as we get by first rotating about the origin and then translating.  (d) To find the matrix that translate Y back to X, we use the concept   Y=ABX

(a) The new coordinates geometrically if X represents the point (0, sqrt{2}) m and this point is rotated about the origin 45^{circ} clockwise and then

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