If $\vec{a}$ and $\vec{b}$ are two non-zero orthogonal vectors, then $|\vec{a} + \vec{b}|$ is equal to

Two vectors $\vec{a}$ and $\vec{b}$ are non-zero and orthogonal. When two vectors are orthogonal, their dot product is zero: $\vec{a} \cdot \vec{b} = 0$.

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To find the magnitude of the vector sum:

$|\vec{a} + \vec{b}|^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})$

$= \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b}$

$= |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2$

Since the vectors are orthogonal, $\vec{a} \cdot \vec{b} = 0$:

$|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2$

$|\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2}$

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Similarly, for $|\vec{a} - \vec{b}|$:

$|\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})$

$= |\vec{a}|^2 - 2\vec{a} \cdot \vec{b} + |\vec{b}|^2$

Since $\vec{a} \cdot \vec{b} = 0$:

$|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2$

$|\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2}$

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Comparing the results:

$|\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2} = |\vec{a} - \vec{b}|$

Therefore, $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$