Question

How would you expand and simplify the following equation?

"(3a + 4b) product dot (5a + 6b)"

Textbook answer: 15| a |^2 + 38a product dot b + 24| b |^2

Answer 1

a is evidently a vector and so is b
in that case I would write a and b as
a = Ax i + Ay j
and
b = Bx i + By j
Then 3 a + 4 b = (3Ax+4Bx)i+(3Ay+4By)j
and 5 a + 6 b = (5Ax+6Bx)i+(5Ay+6By)j
Then the dot product of those two vectors is
(3Ax+4Bx)(5Ax+6Bx)+(3Ay+4By)(5Ay+6By)
which is
(15Ax^2+38AxBx+24Bx^2)+(15Ay^2+38AyBy+24By^2)
which is
15(Ax^2+Ay^2)+38(AxBx+AyBy)+24(Bx^2+By^2)
which is exactly what they said it should be :)