Let's break down each part of the problem one by one.

(a) Inner Product $x_1y_1 + x_2y_2 + x_3y_3$ in terms of a product of $x$ and $y$

This is the definition of the inner product (or dot product) of two vectors $x = (x_1, x_2, x_3)^T$ and $y = (y_1, y_2, y_3)^T$ in $\mathbb{R}^3$. The inner product can be written as:

$x \cdot y = x_1y_1 + x_2y_2 + x_3y_3$

So the expression $x_1y_1 + x_2y_2 + x_3y_3$ is exactly the dot product $x \cdot y$.

(b) Writing $\sum_{i=1}^3 x_i$ in terms of a product of $x$ and another vector

The sum $\sum_{i=1}^3 x_i$ is simply the sum of the components of the vector $x = (x_1, x_2, x_3)^T$. To write this as a product of $x$ and another vector, we can use a vector with all components equal to 1. This is:

$\mathbf{1} = (1, 1, 1)^T$

So the sum of the components of $x$ can be written as:

$\sum_{i=1}^3 x_i = x \cdot \mathbf{1}$

This is the dot product of $x$ with the vector $\mathbf{1} = (1, 1, 1)^T$.

(c) Let $x = (1, 2, 3)^T$ and $y = (1, 1, 1)^T$, calculate $x \cdot y$

We can now calculate the dot product of $x = (1, 2, 3)^T$ and $y = (1, 1, 1)^T$.

$x \cdot y = (1)(1) + (2)(1) + (3)(1)$ $x \cdot y = 1 + 2 + 3 = 6$

Thus, $x \cdot y = 6$.

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Summary:

1. $x_1y_1 + x_2y_2 + x_3y_3$ is the inner product $x \cdot y$.
2. $\sum_{i=1}^3 x_i = x \cdot \mathbf{1}$, where $\mathbf{1} = (1, 1, 1)^T$.
3. The dot product $x \cdot y$ for $x = (1, 2, 3)^T$ and $y = (1, 1, 1)^T$ is 6.

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Would you like more details or have any other questions? Here are some related questions:

1. How do you interpret the geometric meaning of the dot product?
2. What happens if the vectors are orthogonal? How does the dot product behave?
3. How do you generalize this to higher dimensions, like $\mathbb{R}^n$?
4. What is the relationship between the dot product and the cosine of the angle between vectors?
5. How can you compute the dot product in programming languages like Python or MATLAB?

Tip: The dot product is not just a way of multiplying vectors. It also gives you a measure of how aligned two vectors are, which is useful in various fields like physics and computer science!