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The angle between the vectors \(\hat{i}-\hat{j}\) and \(2\hat{i}+3\hat{j}\) is:

1. \(\cos\theta = \frac{1}{5}\)
2. \(\sin\theta = \frac{25}{26}\)
3. \(\tan\theta = 5\)
4. none of these

Correct answer: \(\sin\theta = \frac{25}{26}\)

Solution

For \(\vec{a}=\hat{i}-\hat{j}\) and \(\vec{b}=2\hat{i}+3\hat{j}\), the dot product is \(1\cdot2+(-1)\cdot3=-1\). Their magnitudes are \(\sqrt{2}\) and \(\sqrt{13}\), so \(\cos\theta=-\frac{1}{\sqrt{26}}\). Hence the given options do not match the correct value, so the intended answer is not among the listed choices.

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