48. a) x · x b) x · x · x c) (x · x) · x d) (x · x)x e) (x + 2y) · y f) Ax

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

Explore how to solve problem 48 involving vector dot products and matrix-vector multiplication. Learn key concepts like dot products, scalar multiplication, and matrix operations with a detailed step-by-step guide. Perfect for students in Grades 11-12.

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