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

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 :)

To expand and simplify the equation "(3a + 4b) product dot (5a + 6b)", you can use the distributive property and the properties of the dot product. The dot product of two vectors is calculated by multiplying their corresponding components and then summing them up.

Let's break down the steps to expand and simplify the equation:

Step 1: Expanding the equation
First, use the distributive property to multiply each term in the first vector by each term in the second vector.

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

Step 2: Applying the dot product rule
Now, use the dot product rule to calculate each term.

(3a) dot (5a + 6b) = (3a) dot (5a) + (3a) dot (6b)
(4b) dot (5a + 6b) = (4b) dot (5a) + (4b) dot (6b)

Step 3: Calculate the dot product for each term
Apply the dot product rule, which means multiplying each pair of corresponding terms and summing them up:

(3a) dot (5a) = 3a * 5a = 15| a |^2 (a scalar value)
(3a) dot (6b) = 3a * 6b = 18a product dot b (a vector value)
(4b) dot (5a) = 4b * 5a = 20a product dot b (a vector value)
(4b) dot (6b) = 4b * 6b = 24| b |^2 (a scalar value)

Step 4: Combine the results
Combine all the results from the previous steps to get the simplified equation:

15| a |^2 + 18a product dot b + 20a product dot b + 24| b |^2

Step 5: Simplify further
Combine like terms:

Final simplified equation:
15| a |^2 + 38a product dot b + 24| b |^2

So, the expanded and simplified equation is: 15| a |^2 + 38a product dot b + 24| b |^2.