A vector αi^+βj^ is obtained by rotating 3i^+j^ by 45∘ counterclockwise. The area of the triangle with vertices (α,β),(0,β),(0,0) is:
Solutions · 1
Official solution
3i^+j^ makes angle 30∘ with the x-axis (magnitude 2). After rotating by 45∘, the new vector makes 75∘: (α,β)=2(cos75∘,sin75∘).
The triangle with vertices (α,β),(0,β),(0,0) is right-angled with legs α and β:
\[\text{Area} = \frac12\alpha\beta = \frac12(2\cos75^{\circ})(2\sin75^{\circ}) = \sin150^{\circ} = \frac12\]