Mathematics

Mathematics

Problem

The vectors a and b are such that ∣a∣=4|\mathbf{a}| = 4 , ∣b∣=5|\mathbf{b}| = 5 and the acute angle between a and b is 30∘30^\circ . Find the magnitude of a×b\mathbf{a} \times \mathbf{b} .
Intuitive explanation

The cross product's magnitude depends on how much two vectors "spread apart" in 3D space—it's captured by the sine of the angle between them, not the cosine. Use the formula ∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta, which directly connects the lengths of your vectors and the angle between them to find the area of the parallelogram they span.

Intuitive explanation

The cross product's magnitude depends on how much two vectors "spread apart" in 3D space—it's captured by the sine of the angle between them, not the cosine. Use the formula ∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta, which directly connects the lengths of your vectors and the angle between them to find the area of the parallelogram they span.

Browse the problem library