Question

What is the composition of the translations (T 〈−3, 4〉∘ T 〈8, −7〉)(x, y) as one translation?

Answer 1

Respuesta:

The composition of the translations; T < − 3 ,  4 >  ⋅  T < 8 , − 7 >   (x, y) as one translation gives;

T < 5 ,  − 3 >   (x, y)

    We want to compose the translations; T < − 3 ,  4 > ⋅  T < 8 , − 7 >   (x, y) as one translation;

Now, translation is usually a map defined as;

T < a ,  b>  < x ,  y> = (x + a, y + b)

Thus, the image of the first translation is given as;

T < 8 , − 7 >   (x, y) = (x + 8, y - 7)

Finally;

T < − 3 ,  4 > ⋅  T < 8 , − 7 >   (x, y) = T < − 3 ,  4 > ⋅ (x + 8, y - 7)

⇒ (x + 8 - 3, y - 7 + 4)

⇒ (x + 5, y - 3)

Since T < a ,  b>  < x ,  y> = (x + a, y + b)

Then; (a, b) = (5, -3)

As one translation gives;

T < 5 ,  − 3 >   (x, y)