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The dot product is a fundamental operation in vector mathematics, used extensively to determine the relationship between two vectors. When we talk about the dot product, often represented as \( \mathbf{a} \cdot \mathbf{b} \) for vectors \( \mathbf{a} \) and \( \mathbf{b} \), we're essentially measuring how much of one vector goes in the direction of another.
The formula for the dot product of vectors \( \mathbf{a} = (a_1, a_2, ..., a_n) \) and \( \mathbf{b} = (b_1, b_2, ..., b_n) \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n \)

If the result is zero, it shows that the vectors are perpendicular, as none of the vector's component projects onto the other.
This property plays a crucial role in the problem of proving perpendicularity between the sum and difference of two vectors with the same magnitude.

Vector magnitude refers to the length or size of a vector. For any vector \( \mathbf{v} = (v_1, v_2, ..., v_n) \), its magnitude is calculated using the square root of the sum of the squares of its components:

• \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2} \)

In the provided exercise, both vectors \( \mathbf{a} \) and \( \mathbf{b} \) have the same magnitude, i.e., \( \|\mathbf{a}\| = \|\mathbf{b}\| \).
This is significant because it simplifies the mathematical manipulation during the solution process, ensuring that terms like \( \|\mathbf{a}\|^2 \) and \( \|\mathbf{b}\|^2 \) cancel each other.
This ultimately leads to simplifying the dot product calculation, confirming the perpendicularity requirement.

Perpendicular vectors have a special relationship: their dot product equals zero. This relationship arises because when two vectors are perpendicular, they don't "align" with each other at all, meaning one vector doesn't project onto the other.
In our problem, we are given that vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) need to be shown as perpendicular
To confirm they are perpendicular, we calculate their dot product:

• \( (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \)

Through this computation, by simplifying and using the properties of vector magnitudes and dot products, we see that the terms cancel out, resulting in zero.
This zero result is the mathematical proof required to conclude that the vectors are indeed perpendicular.