Let's solve the given vector operations step by step, keeping in mind the rules of dot products and matrix-vector multiplication.

48a. $\mathbf{x} \cdot \mathbf{x}$

The dot product of a vector $\mathbf{x} = (x_1, x_2, x_3)$ with itself is:

$\mathbf{x} \cdot \mathbf{x} = x_1^2 + x_2^2 + x_3^2$

This results in a scalar value that represents the squared magnitude (or length) of the vector $\mathbf{x}$.

Thus, $\mathbf{x} \cdot \mathbf{x}$ is the squared length of $\mathbf{x}$.

48b. $\mathbf{x} \cdot \mathbf{x} \cdot \mathbf{x}$

This is nonsense because the dot product is a binary operation that produces a scalar, and you cannot take the dot product of a scalar with a vector. Once $\mathbf{x} \cdot \mathbf{x}$ is computed, it results in a scalar, and you can't dot this scalar with another vector $\mathbf{x}$.

Thus, $\mathbf{x} \cdot \mathbf{x} \cdot \mathbf{x}$ is nonsense.

48c. $(\mathbf{x} \cdot \mathbf{x}) \cdot \mathbf{x}$

This is also nonsense. The dot product $\mathbf{x} \cdot \mathbf{x}$ results in a scalar, and you cannot take the dot product of a scalar with a vector.

Thus, $(\mathbf{x} \cdot \mathbf{x}) \cdot \mathbf{x}$ is nonsense.

48d. $(\mathbf{x} \cdot \mathbf{x}) \mathbf{x}$

Here, $\mathbf{x} \cdot \mathbf{x}$ results in a scalar (let's call it $s = \mathbf{x} \cdot \mathbf{x}$), so this expression is the scalar multiplication of vector $\mathbf{x}$ by $s$:

$(\mathbf{x} \cdot \mathbf{x}) \mathbf{x} = s \mathbf{x} = (x_1^2 + x_2^2 + x_3^2)\mathbf{x}$

This operation scales the vector $\mathbf{x}$ by the scalar $s$, which is the squared magnitude of $\mathbf{x}$.

Thus, $(\mathbf{x} \cdot \mathbf{x}) \mathbf{x}$ is a vector in the same direction as $\mathbf{x}$, scaled by the squared magnitude of $\mathbf{x}$.

48e. ( (\mathbf{x} + 2\mathbf{y}) \cd