For which of the following pairs of vectors is the angle between the two vectors obtuse?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2\hat{\imath}-\hat{\jmath}+\hat{k}\) and \(\hat{\imath}+3\hat{\jmath}\)
Since \(\cos\theta = \displaystyle\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}\) and the magnitudes are positive, the angle is obtuse exactly when the scalar product is negative.
\((2)(1)+(-1)(3)+(1)(0) = 2-3 = -1 \lt 0\)
So the angle between \(2\hat{\imath}-\hat{\jmath}+\hat{k}\) and \(\hat{\imath}+3\hat{\jmath}\) is obtuse.
Why the other options are incorrect
Option (1) \(\hat{\imath}+\hat{\jmath}\) and \(\hat{\imath}-\hat{\jmath}+\hat{k}\) — Incorrect. The scalar product is \(1-1+0 = 0\), so the angle is a right angle.
Option (2) \(\hat{\imath}-\hat{\jmath}+2\hat{k}\) and \(2\hat{\imath}+\hat{\jmath}+\hat{k}\) — Incorrect. The scalar product is \(2-1+2 = 3 \gt 0\), so the angle is acute.
Option (4) \(3\hat{\imath}+\hat{k}\) and \(-\hat{\imath}+2\hat{\jmath}+4\hat{k}\) — Incorrect. The scalar product is \(-3+0+4 = 1 \gt 0\), so the angle is acute.