Q: Given vectors $ \mathbf{u} = \langle 3, -2 \rangle $ and $ \mathbf{v} = \langle -1, 4 \rangle $, demonstrate how vector subtraction can be understood as adding the additive inverse. Calculate $ \mathbf{u} - \mathbf{v} $ and explain each step of your solution.

To find $ \mathbf{u} - \mathbf{v} $, we take the vector $ \mathbf{u} = \langle 3, -2 \rangle $ and add the additive inverse of $ \mathbf{v} = \langle -1, 4 \rangle $. The additive inverse of $ \mathbf{v} $ is $ -\mathbf{v} = \langle 1, -4 \rangle $. Thus, $ \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}) = \langle 3, -2 \rangle + \langle 1, -4 \rangle = \langle 3+1, -2-4 \rangle = \langle 4, -6 \rangle $.

Question 1

Construct a well-developed argument that prioritizes the causes and events leading to the Civil War. Be sure to include perspectives from multiple sources, such as political leaders, abolitionists, and Southern landowners. Explain how these perspectives contributed to the different priorities and tensions that ultimately resulted in the Civil War.

Accepted Answer

A comprehensive argument should include various causes and events such as the economic differences between the North and South, the impact of the abolitionist movement, and political compromises like the Missouri Compromise and the Kansas-Nebraska Act. Perspectives from political leaders might focus on states' rights and federal authority, while abolitionists would highlight the moral imperative to end slavery. Southern landowners may prioritize the preservation of their economic system based on slavery. Each perspective contributed to a growing divide, with differing priorities leading to increased tensions and ultimately, the outbreak of the Civil War.

Question 2

When considering a local budget issue, why is it important to seek the perspectives of multiple stakeholders?

Accepted Answer

B) It helps to understand different needs and priorities within the community.

Question 3

Given vectors $ \mathbf{u} = \langle 3, -2 \rangle $ and $ \mathbf{v} = \langle -1, 4 \rangle $, demonstrate how vector subtraction can be understood as adding the additive inverse. Calculate $ \mathbf{u} - \mathbf{v} $ and explain each step of your solution.

Accepted Answer

To find $ \mathbf{u} - \mathbf{v} $, we take the vector $ \mathbf{u} = \langle 3, -2 \rangle $ and add the additive inverse of $ \mathbf{v} = \langle -1, 4 \rangle $. The additive inverse of $ \mathbf{v} $ is $ -\mathbf{v} = \langle 1, -4 \rangle $. Thus, $ \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}) = \langle 3, -2 \rangle + \langle 1, -4 \rangle = \langle 3+1, -2-4 \rangle = \langle 4, -6 \rangle $.

Question 4

Estimate the value of the square root of 50 and explain how you determined your estimate. Then, place it on a number line between two consecutive whole numbers.

Accepted Answer

The square root of 50 is approximately 7.07. To estimate, recognize that 49 is a perfect square, and its square root is 7. Since 50 is slightly more than 49, the square root of 50 will be slightly more than 7. On a number line, $ \sqrt{50} $ is between 7 and 8, closer to 7.

Question 5

How did the development of tools and technology impact early agrarian societies compared to hunter-gatherer societies?

Accepted Answer

C) It enabled the cultivation of crops, leading to permanent settlements and population growth.

Question 6

Explain one way in which a specific treaty changed America's relationship with another nation.

Accepted Answer

The Treaty of Paris in 1783 ended the American Revolutionary War and recognized American independence, significantly changing the relationship between the United States and Great Britain from that of colonies and empire to two separate nations.

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