Find $\mathbf { u } \cdot \mathbf { v }$ And the Angle Between

question 81

Essay

Find $\mathbf { u } \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = 2 \mathbf { i } + 2 \mathbf { j } , \mathbf { v } = - 4 \mathbf { i } - 4 \mathbf { j }$ .

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Definitions:

Industrialization

The transformation from agrarian societies to industrial economies, characterized by the development of manufacturing, technological advancement, and urban growth.

Non-animate Sources

Objects or entities that do not possess life or consciousness, often referred to in discussions about energy, matter, or information.

Fossil Fuels

Natural fuels such as coal, oil, and natural gas, formed from the decomposed remains of plants and animals over millions of years, and used for energy.

Service Industries

Service industries are businesses that provide intangible products or services to consumers, such as healthcare, education, and financial services.