Geometric Translation: Inverse Translations & Translation Vectors

The original translation moves points by ⟨5, 3⟩ because 8 – 3 = 5 and 1 – (–2) = 3, so its inverse must subtract 5 from x and subtract 3 from y to reverse the motion completely.

Because ⟨0, 0⟩ represents no movement at all, reversing it requires another zero movement, making its inverse itself and demonstrating that the identity vector is self-inverse.

To return the figure to its original position, every movement must be undone by applying equal and opposite displacements, which only the vector ⟨–a, –b⟩ accomplishes.

Reversing a translation simply reverses each coordinate adjustment, so undoing a shift of +7 in x and –4 in y requires subtracting 7 from x and adding 4 to y to restore each point’s original location.

Since the original vector moves points 12 units downward, the inverse must move points 12 units upward, reversing the vertical displacement while leaving the horizontal motion unchanged at zero.

The net movement becomes ⟨a + –a, b + –b⟩ = ⟨0, 0⟩, meaning the figure’s final position is identical to its original position, producing no visible change whatsoever.

Adding these vectors yields (4 + –4, –6 + 6) = (0, 0), showing they are perfect opposites and therefore move a point out and then back along the same path with no net displacement.

Inverse translations undo the movement of the original translation without changing shape, size, or orientation because translations are rigid motions, so applying one after the other retraces the path but does not distort the figure.

Since left corresponds to –9 and up corresponds to +4, reversing the translation requires shifting in the opposite directions—right 9 and down 4—to return the figure to its starting position.

Undoing a translation requires applying the exact opposite shifts, meaning any added horizontal or vertical movement must be subtracted to send each point back to its initial coordinates.

Inverse translations undo a movement by applying the same distance in the opposite direction. For a vector ⟨a, b⟩, the inverse is ⟨–a, –b⟩.

Adding the vectors gives (2 + –2, –3 + 3) = (0, 0), meaning their combined effect is zero displacement, so the overall transformation is the identity and the figure does not move at all.

Because the inverse applies the opposite horizontal and vertical displacements, it undoes the effect of ⟨a, b⟩ by retracing the path in reverse, leaving the figure exactly where it started without altering its size, shape, or orientation.

Vectors and their inverses must point in opposite directions with equal magnitude, so reversing a shift of 4 units right and 2 units down requires moving 4 units left and 2 units up to exactly retrace the original motion.

Performing a translation followed by the exact opposite translation cancels the movement because each coordinate change is undone by an equal and opposite change, causing the point to end exactly where it began.

The pair (x, y) → (x – 2, y + 5) and (x, y) → (x + 2, y – 5) is inverse because the second rule exactly cancels the first by shifting right what was shifted left and shifting down what was shifted up, returning every point to its original coordinates.

A translation is undone by performing the opposite movement in each direction, so adding 5 to x reverses the original leftward shift of 5 and subtracting 2 from y reverses the original upward shift of 2, restoring every point to its starting location.

Translation ⟨–3, 7⟩ → M′(1, 2). Applying ⟨3, –7⟩ brings it back: (1 + 3, 2 – 7) = (4, –5).

Inverse translations restore the original coordinates.

Reverse both directions of the vector: ⟨–6, 3⟩ → ⟨6, –3⟩. Positive x means right, and negative y means down.

To reverse a translation, switch the signs of both components: +4 → –4, –7 → +7. The inverse rule becomes (x, y) → (x – 4, y + 7).