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.