Evaluate the product .
The dot product of ( 3 a → − 5 b → ) ⋅ ( 2 a → + 7 b → ) is,
( 3 a → − 5 b → ) ⋅ ( 2 a → + 7 b → ) = 3 a → ( 2 a → + 7 b → ) − 5 b → ( 2 a → + 7 b → ) = 3 a → ⋅ 2 a → + 3 a → ⋅ 7 b → − 5 b → ⋅ 2 a → − 5 b → ⋅ 7 b → = 3 a → ⋅ 2 a → + 21 a → ⋅ b → − 10 b → ⋅ a → − 35 b → ⋅ b → = 6 a → ⋅ a → + 11 a → ⋅ b → − 35 b → ⋅ b →
We know that a → ⋅ a → = | a → | 2 and b → ⋅ b → = | b → | 2 .
Now, further simplify,
6 a → ⋅ a → + 11 a → ⋅ b → − 35 b → ⋅ b → = 6 | a → | 2 + 11 a → ⋅ b → − 35 | b → | 2
Thus, the product of ( 3 a → − 5 b → ) ⋅ ( 2 a → + 7 b → ) is 6 | a → | 2 + 11 a → ⋅ b → − 35 | b → | 2 .
The dot product of
We know that
Now, further simplify,
Thus, the product of
