Evaluate the product (3a⃗ - 5b⃗) ⋅ (2a⃗ + 7b⃗).
6|a⃗|² + 11(a⃗ · b⃗) - 35|b⃗|²
Explanation
The question requires the algebraic evaluation of a scalar product. Since specific vectors are not provided, the answer is derived by expanding the expression using distributive and scalar multiplication properties found in the context. The context explicitly provides Property 2 for scalar multiplication and demonstrates the distributive property in the expansion of |a⃗ + b⃗ + c⃗|². It also establishes that a⃗ · a⃗ = |a⃗|² and implies commutativity (a⃗ · b⃗ = b⃗ · a⃗) through the symmetric nature of the component form expansion.
Solution Steps
Step 1: Expand the product using the distributive property of the scalar product over addition: (3a⃗ - 5b⃗) ⋅ (2a⃗ + 7b⃗) = 3a⃗ ⋅ 2a⃗ + 3a⃗ ⋅ 7b⃗ - 5b⃗ ⋅ 2a⃗ - 5b⃗ ⋅ 7b⃗.
Step 2: Apply Property 2 (scalar multiplication) to take out the scalar coefficients: = 6(a⃗ ⋅ a⃗) + 21(a⃗ ⋅ b⃗) - 10(b⃗ ⋅ a⃗) - 35(b⃗ ⋅ b⃗).
Step 3: Simplify using the property a⃗ ⋅ a⃗ = |a⃗|² and the commutative property (a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗): = 6|a⃗|² + 21(a⃗ ⋅ b⃗) - 10(a⃗ ⋅ b⃗) - 35|b⃗|².
Step 4: Combine the like terms to reach the final answer: = 6|a⃗|² + 11(a⃗ ⋅ b⃗) - 35|b⃗|².