To solve this problem, we need to determine what the equality \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \) implies about the vectors \( \vec{a} \) and \( \vec{b} \). We are given:
\(\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}\),
\(\vec{b} = \alpha\hat{i} + \beta\hat{j} + 2\hat{k}\).
First, calculate \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\):
\(\vec{a} + \vec{b} = (2 + \alpha)\hat{i} + (2 + \beta)\hat{j} + (-1 + 2)\hat{k}\)
\(\vec{a} + \vec{b} = (\alpha + 2)\hat{i} + (\beta + 2)\hat{j} + \hat{k}\).
\(\vec{a} - \vec{b} = (2 - \alpha)\hat{i} + (2 - \beta)\hat{j} + (-1 - 2)\hat{k}\)
\(\vec{a} - \vec{b} = (2 - \alpha)\hat{i} + (2 - \beta)\hat{j} - 3\hat{k}\).
The magnitudes are:
\(|\vec{a} + \vec{b}| = \sqrt{(\alpha + 2)^2 + (\beta + 2)^2 + 1^2}\),
\(|\vec{a} - \vec{b}| = \sqrt{(2 - \alpha)^2 + (2 - \beta)^2 + (-3)^2}\).
Given that these magnitudes are equal, we set up the equation:
\[ (\alpha + 2)^2 + (\beta + 2)^2 + 1^2 = (2 - \alpha)^2 + (2 - \beta)^2 + (-3)^2 \]Simplify both sides:
The left side becomes:
\[ (\alpha + 2)^2 + (\beta + 2)^2 + 1 = \alpha^2 + 4\alpha + 4 + \beta^2 + 4\beta + 4 + 1 \] \[ = \alpha^2 + \beta^2 + 4\alpha + 4\beta + 9 \]The right side becomes:
\[ (2 - \alpha)^2 + (2 - \beta)^2 + 9 = \alpha^2 - 4\alpha + 4 + \beta^2 - 4\beta + 4 + 9 \] \[ = \alpha^2 + \beta^2 - 4\alpha - 4\beta + 17 \]Equate the two expressions:
\[ \alpha^2 + \beta^2 + 4\alpha + 4\beta + 9 = \alpha^2 + \beta^2 - 4\alpha - 4\beta + 17 \]Cancel common terms and simplify:
\[ 8\alpha + 8\beta = 8 \]Divide through by 8:
\[ \alpha + \beta = 1 \]Thus, the value of \(\alpha + \beta\) is 1. Therefore, the correct answer is: