Final answer:
To find u · v, multiply the corresponding components of the vectors and sum the products. To find v · v, multiply the corresponding components of the vector and sum the products. To find u², square each component of the vector. To find (u · v)v, find u · v and multiply each component of the vector v by the result. To find u · (5v), multiply each component of the vector v by 5 and find the dot product with u.
Step-by-step explanation:
To find u · v, we multiply the corresponding components of the vectors and sum the products. So, u · v = (3)(4) + (-3)(2) = 12 - 6 = 6.
To find v · v, we again multiply the corresponding components of the vector and sum the products. So, v · v = (4)(4) + (2)(2) = 16 + 4 = 20.
To find u², we square each component of the vector. So, u² = (3)² + (-3)² = 9 + 9 = 18.
To find (u ·v)v, we first find u · v using the previous calculation (u · v = 6), and then multiply each component of the vector v by the result. So, (u · v)v = (6)(4, 2) = (24, 12).
To find u · (5v), we first multiply each component of the vector v by 5 and then find the dot product with u. So, u · (5v) = (3)(20, 10) + (-3)(10, 5) = (60 - 30) + (-30 - 15) = 30 - 45 = -15.