Translations
When you press "3 to the right and 2 up" in a video game, the character is translated: all of its points move the same amount, in the same direction. That instruction "3 to the right and 2 up" is a vector, and vectors are used to describe every translation.
Vectors
Look at the grid, in which each side of a small square measures one unit, and determine the components of the translation vector $\vec{u}$ that moves point $P$ to the…
Look at the grid, in which each side of a small square measures one unit, and determine the components of the translation vector $\vec{u}$ that moves point $P$ to the position of point $P'$.
Identify which of the following options correctly describes a translation: a) Turning a key in a lock. b) The movement of a drawer when opening it. c) The reflection of…
Identify which of the following options correctly describes a translation:
- a) Turning a key in a lock.
- b) The movement of a drawer when opening it.
- c) The reflection of a mountain in a lake.
- d) The movement of the hands of a clock.
Translating points and shapes
Move the vector and watch: all the points move by the same amount, and the shape neither turns nor becomes distorted.
Given the line segment with endpoints $A(-3, 1)$ and $B(1, 2)$, calculate the coordinates of the transformed segment $A'B'$ after applying a translation with vector…
Given the line segment with endpoints $A(-3, 1)$ and $B(1, 2)$, calculate the coordinates of the transformed segment $A'B'$ after applying a translation with vector $\vec{v} = (4, 3)$. Plot both line segments on a set of coordinate axes.
Apply a translation with vector $\vec{v} = (6, 2)$ to the triangle with vertices $V_1(0, 0)$, $V_2(2, 0)$ and $V_3(1, 3)$. 1. Write the coordinates of the new vertices.…
Apply a translation with vector $\vec{v} = (6, 2)$ to the triangle with vertices $V_1(0, 0)$, $V_2(2, 0)$ and $V_3(1, 3)$.
-
Write the coordinates of the new vertices.
-
How does the area of the new triangle compare to the original one? Give reasons for your answer.
If we translate point $Q$ by vector $\vec{v} = (-4, 7)$ we obtain point $Q'(0, 2)$. What were the coordinates of the original point $Q$ before the translation?
If we translate point $Q$ by vector $\vec{v} = (-4, 7)$ we obtain point $Q'(0, 2)$. What were the coordinates of the original point $Q$ before the translation?
Chaining translations: adding vectors
Applying a translation by vector $\vec u$ and then another by vector $\vec v$ is the same as a single one by vector $\vec u + \vec v$:
$$\vec u + \vec v = (u_1 + v_1,\ u_2 + v_2)$$And the inverse translation of $\vec v$ is the one by vector $-\vec v$: it undoes the movement.
An object undergoes two consecutive translations. The first is defined by the vector $\vec{u} = (2, 5)$ and the second by the vector $\vec{v} = (3, -1)$. What is the…
An object undergoes two consecutive translations. The first is defined by the vector $\vec{u} = (2, 5)$ and the second by the vector $\vec{v} = (3, -1)$. What is the vector of the single translation equivalent to carrying out the previous two one after the other?
Two translations are said to be inverse if, when applied one after the other, the final point coincides with the initial point (yielding the identity). a) If an object…
Two translations are said to be inverse if, when applied one after the other, the final point coincides with the initial point (yielding the identity).
- a) If an object has been translated by vector $\vec{v} = (-7, 4)$, what is the vector of the inverse translation?
- b) If we apply translation $\vec{u} = (3, -2)$ to point $M(x, y)$ and then its inverse, what are the coordinates of the resulting point?
- c) A boat travels from port $A(10, 20)$ to a buoy $B$ via vector $\vec{v} = (50, -30)$. Due to a storm, it must return immediately to port. What translation vector must it follow to return exactly to point $A$?
A warehouse robot moves along the floor of the building following a straight path. The technician has defined its movement using the composition of three consecutive…
A warehouse robot moves along the floor of the building following a straight path. The technician has defined its movement using the composition of three consecutive translations:
-
$\vec{v_1} = (5, -2)$
-
$\vec{v_2} = (-3, 7)$
-
$\vec{v_3} = (a, b)$
If the robot starts at point $(0, 0)$ and after the three movements ends up at point $(4, 4)$, what must the components of the third vector $\vec{v_3}$ be?
Back to the start: the instruction "3 to the right and 2 up" is represented by the vector $\vec{v} = (3, 2)$, which moves any point $(x, y)$ to $(x + 3, y + 2)$ along a straight-line path of $|\vec{v}| = \sqrt{3^2 + 2^2} = \sqrt{13} \approx 3{,}6$ squares.