Let \(\vec{a}=\hat{\imath}+2\hat{\jmath}-\hat{k}\), \(\vec{b}=2\hat{\imath}-\hat{\jmath}+\hat{k}\) and \(\vec{c}=7\hat{\imath}+4\hat{\jmath}-\hat{k}\). If \(\vec{c} = x\vec{a}+y\vec{b}\), then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(x = 3,\ y = 2\)
Writing out the components:
\(x\vec{a}+y\vec{b} = (x+2y)\hat{\imath}+(2x-y)\hat{\jmath}+(-x+y)\hat{k}\)
Two vectors are equal only when their corresponding components are equal:
\(x+2y = 7\)
\(2x-y = 4\)
\(-x+y = -1\)
Adding twice the second equation to the first:
\(5x = 15\)
\(x = 3, \quad y = 2\)
These values also satisfy the third equation: \(-3+2 = -1\).