Every point moves the same way
A translation slides a shape without turning it. Every point of the shape moves the same distance in the same direction.
Slide the triangle with corners (1, 2), (3, 2) and (3, 4) 3 squares to the right and 1 square down. Every corner makes that same move: (1, 2) goes to (4, 1), (3, 2) goes to (6, 1), and (3, 4) goes to (6, 3).
The triangle and its image, dashed, 3 to the right and 1 down. The gold arrow is the move every corner makes, drawn from one corner.
The column vector
The move is written as a column vector: two numbers stacked in brackets, the move across on top and the move up underneath. Across is positive to the right and negative to the left; up is positive upward and negative downward. Written on one line, the move 3 right and 1 down is [3, −1].
To translate a point by a vector, add the top number to the x-coordinate and the bottom number to the y-coordinate: (x, y) moved by [a, b] lands on (x + a, y + b). From (1, 2), the vector [3, −1] gives (1 + 3, 2 + (−1)) = (4, 1).
A point and a vector are different things. The point (1, 2) is a position. The vector [3, −1] is a move, and the same move is made from every corner.
Finding the vector
When a point and its image are given, the vector is the image minus the point, one coordinate at a time. (1, 2) goes to (4, 1), so the vector is [4 − 1, 1 − 2] = [3, −1].
The order matters: image minus object. Subtracting the other way round, (1, 2) − (4, 1), gives [−3, 1], the move from the image back to the object.
The same shape, facing the same way
Both ends of every side move by the same vector, so every side keeps its length and its direction. The bottom side of the triangle runs from (1, 2) to (3, 2), 2 squares long and horizontal. Its image runs from (4, 1) to (6, 1), still 2 squares long and still horizontal.
Because every side is parallel to its image, nothing turns: the image faces the same way as the object. And because every length is the same, nothing stretches. The image is congruent to the object, in a new position. This is the difference from a reflection, which turns a shape over, and a rotation, which turns it round.
The object A and its image A' after the translation [3, −1]. Each side of A' is parallel to, and as long as, the matching side of A.
Two translations in a row
Translate by [3, −1], and then translate the image by [−2, 4]. A point (x, y) goes first to (x + 3, y − 1), and then to (x + 3 − 2, y − 1 + 4) = (x + 1, y + 3).
So the two moves together are one translation, by [1, 3], and [1, 3] = [3 + (−2), −1 + 4]. Two translations in a row add their vectors: add the top numbers, and add the bottom numbers. Check with (1, 2): the first move takes it to (4, 1), and the second to (2, 5), which is (1 + 1, 2 + 3).
To undo a translation, move by the same amounts the other way. The vector back is the negative of the vector: [3, −1] is undone by [−3, 1], and the two add to [0, 0], no move at all.
b = (−1, 2)
Make the resultant lie along the y-axis
The white arrow a = (3, 1) is a first translation, 3 right and 1 up, and the dashed arrow b is a second, starting where the first ends; the figure writes each vector in a row. The gold arrow, called the resultant, is the single translation that does both: here (3, 1) + (−1, 2) = (2, 3). Drag the head of b until the resultant points straight up: its across part must cancel the 3, so b must be 3 to the left.
The usual mistakes
Ignoring a minus sign. [4, −2] means 2 down, so (2, 3) lands on (6, 1), not (6, 5).
Subtracting the vector instead of adding it. (2, 3) − [4, −2] = (−2, 5) runs the move backwards; a translation by [4, −2] adds it.
Reading the numbers in the wrong order. The top number, written first, is the move across, and the bottom number is the move up.
Thinking a translation changes the shape. Only the position changes; the size, the angles and the way the shape faces all stay the same.
Worked example: A Sofa Moved Twice on a Floor Plan, and the Vector That Brings It Back
Question On a floor plan marked in meters, a sofa is the rectangle with corners A(1, 1), B(4, 1), C(4, 2) and D(1, 2). It is slid across the floor by the vector 52, and then, after a rug has been laid, by the vector −23. (a) Find the coordinates of the corners of the sofa after both moves. (b) Find the single vector that would have made both moves at once, and the vector that would slide the sofa straight back to where it started.
1.The first move adds 5 to every x-coordinate and 2 to every y-coordinate. So A(1, 1) goes to (1 + 5, 1 + 2) = (6, 3), B(4, 1) goes to (9, 3), C(4, 2) goes to (9, 4) and D(1, 2) goes to (6, 4).
The first move adds 5 to each x-coordinate and 2 to each y-coordinate: A(1, 1) goes to A'(6, 3). 2.The second move subtracts 2 from every x-coordinate and adds 3 to every y-coordinate. So (6, 3) goes to (6 − 2, 3 + 3) = (4, 6), (9, 3) goes to (7, 6), (9, 4) goes to (7, 7) and (6, 4) goes to (4, 7).
The second move, −23, takes A'(6, 3) to A''(4, 6), and every other corner with it. 3.(a) After both moves the corners are at (4, 6), (7, 6), (7, 7) and (4, 7). The sofa is still 3 m long and 1 m deep, because a translation keeps every length.
(a) After both moves the corners are (4, 6), (7, 6), (7, 7) and (4, 7), and the sofa is still 3 m by 1 m. 4.One move that does the work of both is found by adding the two vectors: 52 + −23 = 5 − 22 + 3 = 35. Check: A(1, 1) moved by this vector lands at (4, 6), the corner found in two moves.
One move that does both is the sum: 52 + −23 = 35. 5.(b) The single vector is 35. To slide the sofa straight back, every point must move 3 m in the negative x-direction and 5 m in the negative y-direction, so the vector back is −3−5. Check: C''(7, 7) moved by it lands at (4, 2), where C started.
(b) The single vector is 35, and the vector back is −3−5: it takes C''(7, 7) back to C(4, 2).
Answer: (a) (4, 6), (7, 6), (7, 7) and (4, 7); (b) the single vector is 35, and the vector back is −3−5
Common mistakes
- Subtracting the vector instead of adding it, which sends A to (1 − 5, 1 − 2) = (−4, −1), outside the room. The top number of the vector is how far the sofa moves in the positive x-direction, so it is added to each x-coordinate.
- Giving the way back as the second vector with its signs changed, 2−3. That undoes only the second move and leaves the sofa where the first move put it; the way back must undo the whole of 35.
Worked example: A Hiking Map on a Tablet: the Slide That Puts the Campsite in the Middle of the Window
Question A hiking app shows a rectangular window of a map. On the map's grid, marked in kilometers, the window has corners (0, 0), (6, 0), (6, 4) and (0, 4). A hiker taps a campsite at (10, 7), and the app slides the window across the map, without turning it or changing its size, until the campsite is at the center of the window. (a) Find the vector by which the window is translated. (b) Find the corners of the window after the slide, and decide whether a waterfall at (14, 6) can be seen in the window.
1.The center of the window is halfway along each side: x = 0 + 62 = 3 and y = 0 + 42 = 2. So the center starts at (3, 2).
The center of the window is halfway along each side, at (3, 2). 2.The center must end at the campsite, (10, 7). The vector from (3, 2) to (10, 7) is 10 − 37 − 2 = 75.
From the center (3, 2) to the campsite (10, 7) is 10 − 37 − 2 = 75. 3.(a) The window is translated by 75, which is 7 km in the x-direction and 5 km in the y-direction.
(a) The window is translated by 75: 7 km in the x-direction and 5 km in the y-direction. 4.Every corner moves by the same vector: (0, 0) goes to (7, 5), (6, 0) goes to (13, 5), (6, 4) goes to (13, 9) and (0, 4) goes to (7, 9). Check: the center of the new window is (7 + 132, 5 + 92) = (10, 7), which is the campsite.
Every corner moves by the same vector, to (7, 5), (13, 5), (13, 9) and (7, 9), and the middle of the new window is the campsite. 5.(b) The new window has corners (7, 5), (13, 5), (13, 9) and (7, 9), so it shows x from 7 to 13 and y from 5 to 9. The waterfall's y-coordinate, 6, is in that range, but its x-coordinate, 14, is 1 km beyond the right-hand edge at x = 13. The waterfall cannot be seen in the window.
(b) The new window runs from x = 7 to x = 13. The waterfall at (14, 6) is 1 km beyond its right-hand edge, so it cannot be seen.
Answer: (a) 75; (b) the corners are (7, 5), (13, 5), (13, 9) and (7, 9), and the waterfall at (14, 6) cannot be seen: it is 1 km beyond the right-hand edge
Common mistakes
- Moving the corner (0, 0) onto the campsite and translating by 107. That puts the campsite at the bottom left corner of the window, not at its center.
- Subtracting the wrong way round and giving −7−5. The vector from the center to the campsite is the campsite's coordinates minus the center's; the reverse vector moves the window away from the campsite.