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The gradient vector, often denoted as \( abla f \), is a vector that provides valuable information about the rate and direction of change of a function at a particular point. It is composed of the partial derivatives of the function with respect to each variable in a multivariable function. For instance, in the case of the function \( f(x, y) \), the gradient vector is defined as:

• \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

The direction of this vector indicates the direction in which the function increases the most rapidly, while its magnitude conveys how steeply the function is climbing. In the original exercise, the gradient vector at the point \((100, 100)\) is given as \(2\vec{i} + 3\vec{j}\), highlighting that the function exhibits more rapid change with respect to \(y\).
The gradient vector is a crucial tool in finding the directional derivative, which tells us how the function changes as we move in a specific direction.

Normalization is a process that converts any given vector into a unit vector.

• A unit vector maintains the same direction as the original vector but has a magnitude of 1.
• This is vital when calculating the directional derivative since the direction must be a unit vector.

To normalize a vector, you divide each component of the vector by its magnitude. Consider the vector \( \vec{i} + \vec{j} \):

• Calculate the magnitude: \( \| \vec{i} + \vec{j} \| = \sqrt{1^2 + 1^2} = \sqrt{2} \).
• Divide each component by this magnitude to create the normalized vector: \( \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \).

Normalization ensures that the directional information aligns correctly with calculations, allowing for accurate determination of how a function changes in that specified direction.

Vector magnitude, or the length of a vector, is a measure of how long the vector is. For a given vector \( \vec{a} = (a_1, a_2) \), its magnitude is calculated as:

• \( \| \vec{a} \| = \sqrt{a_1^2 + a_2^2} \).

This formula is essentially derived from the Pythagorean theorem, applied in a two-dimensional space. Magnitude plays a key role in normalization, where it is utilized to scale a vector to have a length of 1.
For instance, calculating the magnitude of \( \vec{i} + \vec{j} \) reveals \( \sqrt{2} \), enabling us to normalize it effectively.Understanding the magnitude gives insight into the size of the vector, which is a critical factor when examining the overall geometry of vector spaces.

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Essentially, it is a way to multiply two vectors to yield a scalar, and is calculated by summing the products of their corresponding components:

• For vectors \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), the dot product is \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \).

The dot product is particularly significant in calculating the directional derivative, as it quantifies how much one vector extends in the direction of another. In our original problem, the dot product \( (2, 3) \cdot \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \) results in \( \frac{5}{\sqrt{2}} \), which represents the rate of change of the function in the specified direction.
The dot product thus bridges vectors and scalars, providing a fundamental tool to interpret vector alignment and influence on space.