Final answer:
To calculate the dot product of -7u and v, we first find the components of u and multiply by -7, then find the components of v, and finally multiply the corresponding components and add them together, resulting in -21√6 - 21√2.
Step-by-step explanation:
To find the dot product of vectors -7u and v, we first multiply vector u by -7 and then compute the dot product with vector v. The components of u are given by u = -2(cos 30°i + sin 30°j), simplifying this with the exact values of sine and cosine, we have u = -2(√3/2 i + 1/2 j) which is u = -√3 i - j. Therefore, -7u = 7√3 i + 7j.
The components of v are given by v = 6(cos 225°i + sin 225°j). Using the exact values for these trigonometric functions, we get cos 225° = -√2/2 and sin 225° = -√2/2, so v = 6(-√2/2 i - √2/2 j) which simplifies to v = -3√2 i - 3√2 j.
Now, the dot product -7u · v is the sum of the products of the corresponding components of the two vectors. So, (7√3 i + 7j) · (-3√2 i - 3√2 j) is equal to (7√3)(-3√2) + (7)(-3√2), which simplifies to -21√6 - 21√2. This is the final result of the dot product of -7u and v.