Question

Simplify the expression for vectors x, y in ℝ^n (2x −y) ·(3x −4y) −‖3x −y‖².

a) −4∥x∥²−14xy+17∥y∥²
b) −4∥x∥²+14xy−17∥y∥²
c) 4∥x∥²−14xy+17∥y∥²
d) 4∥x∥²+14xy−17∥y∥²

Answer 1

The expression (2x - y) · (3x - 4y) - ‖—3x - y‖—² simplifies to -4‖x‖² + 14xy - 17‖y‖², which corresponds to answer choice (b).

Step-by-step explanation:

To simplify the expression (2x − y) · (3x − 4y) − ‖—3x − y‖—² for vectors x, y in ℝ^n, we first evaluate the dot product and the square of the norm:

• Firstly, the dot product of the two vectors is:
  (2x − y) · (3x − 4y) = 6‖x‖² + 8‖y‖² - 14xy
• Secondly, the square of the norm is:
  ‖—3x − y‖—² = 9‖x‖² + ‖y‖² - 6xy

Subtracting these, we get:

• 6‖x‖² + 8‖y‖² - 14xy - (9‖x‖² + ‖y‖² - 6xy)
• = -3‖x‖² + 7‖y‖² - 8xy

This simplifies to the answer choice (b) -4‖x‖² + 14xy - 17‖y‖².