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provide a step-by-step solution to the problem in the screenshot 

We are asked to determine whether the identity holds:

\[
|a - b|^2 = |a|^2 + |b|^2 - 2(a_x b_x + a_y b_y + a_z b_z)
\]
where \(a, b \in \mathbb{R}^3\).

### Step 1: Recall the formula for the magnitude squared of a vector

For any vector \( v = (v_x, v_y, v_z) \in \mathbb{R}^3 \), its magnitude squared is given by:

\[
|v|^2 = v_x^2 + v_y^2 + v_z^2
\]

This means the magnitude squared of a vector is the sum of the squares of its components.

### Step 2: Expand \( |a - b|^2 \)

The vector difference \( a - b \) has components \( (a_x - b_x, a_y - b_y, a_z - b_z) \), so we can write:

\[
|a - b|^2 = (a_x - b_x)^2 + (a_y - b_y)^2 + (a_z - b_z)^2
\]

Now expand each square:

\[
(a_x - b_x)^2 = a_x^2 - 2a_x b_x + b_x^2
\]
\[
(a_y - b_y)^2 = a_y^2 - 2a_y b_y + b_y^2
\]
\[
(a_z - b_z)^2 = a_z^2 - 2a_z b_z + b_z^2
\]

Thus, we have:

\[
|a - b|^2 = (a_x^2 + a_y^2 + a_z^2) + (b_x^2 + b_y^2 + b_z^2) - 2(a_x b_x + a_y b_y + a_z b_z)
\]

### Step 3: Interpret the terms

Notice that:
- \( a_x^2 + a_y^2 + a_z^2 = |a|^2 \), which is the magnitude squared of vector \( a \),
- \( b_x^2 + b_y^2 + b_z^2 = |b|^2 \), which is the magnitude squared of vector \( b \),
- \( a_x b_x + a_y b_y + a_z b_z = a \cdot b \), which is the dot product of vectors \( a \) and \( b \).

Thus, the expanded form becomes:

\[
|a - b|^2 = |a|^2 + |b|^2 - 2(a \cdot b)
\]

### Step 4: Compare with the given expression

The given expression in the problem is:

\[
|a - b|^2 = |a|^2 + |b|^2 - 2(a_x b_x + a_y b_y + a_z b_z)
\]

This matches exactly with the expanded form of \( |a - b|^2 \), since \( a_x b_x + a_y b_y + a_z b_z \) is the dot product \( a \cdot b \).

### Conclusion:

The identity holds true. Therefore, the expression given in the problem is correct.

---

Would you like to explore more details or have any questions on this? Here are 5 related questions to deepen understanding:

1. What is the geometric interpretation of the dot product \( a \cdot b \)?
2. How does the magnitude squared of \( |a - b| \) relate to the distance between the vectors \( a \) and \( b \)?
3. Can this result be generalized to \( \mathbb{R}^n \) (for vectors in \( n \)-dimensional space)?
4. How would the result change if we considered vectors in \( \mathbb{C}^3 \) (complex space)?
5. What is the physical meaning of the magnitude and dot product in physics, particularly in mechanics?

Tip: Always break down vectors component-wise when working with their magnitudes and dot products—it helps simplify complex expressions.

Let a, b ∈ R³. Determine whether |a - b|² = |a|² + |b|² - 2(axbx + ayby + azbz).

This problem asks to verify the identity |a - b|² = |a|² + |b|² - 2(axbx + ayby + azbz) in R³. The solution involves expanding and interpreting vector algebra expressions using the dot product and magnitude formulas.

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