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Solve the problem provided in the screenshot and provide a step-by-step solution

We are given the vectors \( a = [1 \ 0 \ 0] \) and \( b = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \). We are asked to compute the following:

1. \( a \cdot b \) (the dot product of \( a \) and \( b \))
2. \( b \cdot a \) (the matrix product of \( b \) and \( a \))

### Step 1: Compute \( a \cdot b \) (Dot Product)
The dot product of two vectors is computed as the sum of the products of their corresponding components. 

Given:
\[
a = [1 \ 0 \ 0], \quad b = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}
\]
We compute:
\[
a \cdot b = 1 \times 1 + 0 \times 0 + 0 \times 0 = 1
\]

### Step 2: Compute \( b \cdot a \) (Matrix Product)
The matrix product of a column vector \( b \) and a row vector \( a \) results in a matrix. The matrix product is performed by multiplying each element of \( b \) with each element of \( a \).

We have:
\[
b = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad a = [1 \ 0 \ 0]
\]
The result will be a \( 3 \times 3 \) matrix:
\[
b \cdot a = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \times [1 \ 0 \ 0] = \begin{bmatrix} 1 \times 1 & 1 \times 0 & 1 \times 0 \\ 0 \times 1 & 0 \times 0 & 0 \times 0 \\ 0 \times 1 & 0 \times 0 & 0 \times 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
\]

### Final Answers:
1. \( a \cdot b = 1 \)
2. \( b \cdot a = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)

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Would you like further explanation or have any questions?

Here are five related questions you might find interesting:
1. What is the geometric interpretation of the dot product of two vectors?
2. How does the result change if both vectors \( a \) and \( b \) are unit vectors but not aligned?
3. What happens when you compute the cross product instead of the dot product?
4. Can the dot product ever result in a negative value? Under what conditions?
5. How is the matrix product \( b \cdot a \) different when \( a \) and \( b \) are not perpendicular?

**Tip:** The dot product is useful for finding the angle between two vectors and determining their orthogonality.

Let a = [1 0 0] and b = [1 0 0], compute a · b and b · a.

Learn how to compute the dot product and matrix product for the vectors a = [1 0 0] and b = [1 0 0]. This problem covers key concepts in linear algebra, including dot products and matrix multiplication. Suitable for students in Grades 10-12.

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