Discussion
Consider vectors \(\vec{a} = \hat{i} + 5\hat{j} + \hat{k}\), \(\vec{b} = \hat{i} - 5\hat{j} + \hat{k}\) and \(\vec{c} = \hat{i} + \hat{k}\). Which of the following statements is true ?
Answer: B
The vectors are linearly dependent.
Adding the first two vectors cancels the \(\hat{j}\) parts:
\[ \vec{a} + \vec{b} = 2\hat{i} + 2\hat{k} = 2\vec{c} \]
so \(\vec{a} + \vec{b} - 2\vec{c} = 0\), a non-trivial linear combination that vanishes. Equivalently, the determinant of their components is zero.
They are therefore not independent. They are not orthogonal either: \(\vec{a}\cdot\vec{b} = 1 - 25 + 1 = -23\). And none is normalised, since \(|\vec{a}| = \sqrt{27}\) and \(|\vec{c}| = \sqrt{2}\).
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