Let $\vec { v } = 3 \vec { i } + a \vec { j } + b \vec { k }$

question 40

Essay

Let $\vec { v } = 3 \vec { i } + a \vec { j } + b \vec { k }$ be a vector in space with a, b > 0.
Compute the cross product $\vec { v } \times ( 3 \vec { j } + \vec { k } )$ and then use the result and the Lagrange Multiplier method to find the values of a and b such that the magnitude of the cross product $\| \vec { v } \times ( 3 \vec { j } + \vec { k } ) \|$ is the largest with $\| \vec { v } \| = 19 .$

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Definitions:

Total Market

The entire demand for a product or service within a particular industry, including all potential customers.

Differentiated Product

A product that is distinct from similar products offered by competitors due to unique features, branding, or quality.

Oligopoly

A market structure characterized by a small number of firms controlling a significant portion of the market share, leading to limited competition and often collaborative behavior.

Conditions of Entry

The factors that determine the ease or difficulty with which new firms can enter an industry and compete with existing firms.