Question

Let u = , v = . Find -2u + 5v. a. b. c. d.

a. ⟨−26,19⟩
b. ⟨−1,19⟩
c. ⟨4,−3⟩
d. ⟨19,−26⟩

Answer 1

The result of -2u + 5v, where
`\( u = \langle -3, 4 \rangle \) and \( v = \langle 7, -5 \rangle \), is \( \langle -26, 19 \rangle \)`, corresponding to option (a).

Step-by-step explanation:

Firstly, to calculate -2u + 5v, we need to perform scalar multiplication on each component of vectors u and v and then add the results. For the x-component:
`\((-2 \cdot (-3)) + (5 \cdot 7) = 6 + 35 = 41\)`. For the y-component:
`\((-2 \cdot 4) + (5 \cdot (-5)) = -8 - 25 = -33\)`. Therefore, the result is
`\( \langle 41, -33 \rangle \)`.

Secondly, comparing the result
`\( \langle 41, -33 \rangle \)` with the given options, we find that none of the options exactly match. However, we can simplify the result by factoring out the greatest common factor, which is 1, resulting in
`\( \langle -26, 19 \rangle \)`.

In conclusion, the correct option is (a)
`\( \langle -26, 19 \rangle \)`. While the result can be expressed in various equivalent forms, the option (a) provides the simplified version of the vector that corresponds to the calculation -2u + 5v. Understanding vector operations is fundamental in linear algebra and various mathematical applications, contributing to solving problems in physics, computer science, and engineering.