Question

If \[ \vec{a} = -2\hat{i} + 9\hat{j} - 6\hat{k}, \quad \vec{b} = t\hat{i} - 2\hat{j} + 6\hat{k} \] are vectors such that \(|\vec{a} + \vec{b}| = 25\), then the sum of the values of \(t =\ ?\)

• A. \(14\)
• B. \(11\)
• C. \(\mathbf{4}\)
• D. \(77\)

Hint:

When given magnitude of a vector expression, square both sides and solve the resulting quadratic carefully.

Answer 1

The Correct Option is C

\[ \vec{a} + \vec{b} = (-2 + t)\hat{i} + (9 - 2)\hat{j} + (-6 + 6)\hat{k} = (t - 2)\hat{i} + 7\hat{j} \] Magnitude: \[ |\vec{a} + \vec{b}| = \sqrt{(t - 2)^2 + 7^2} = 25 \Rightarrow (t - 2)^2 + 49 = 625 \Rightarrow (t - 2)^2 = 576 \Rightarrow t - 2 = \pm 24 \Rightarrow t = 26 \text{ or } -22 \] Sum of values: \[ t = 26 + (-22) = \boxed{4} \]