Question

If v × w = ⟨-1, 0, 4⟩ and v ⋅ w = 6, find tan(θ), where θ is the angle between v and w.

Answer 1

To find tan(θ), we can use the formula tan(θ) = sqrt((v × w)² / (v · v)(w · w)). Given that v × w = ⟨-1, 0, 4⟩ and v ⋅ w = 6, we can substitute these values into the formula to get tan(θ) = sqrt(17) / (sqrt(v · v) * sqrt(w · w)). However, without the magnitude of vectors v and w, we cannot determine the exact value of tan(θ).

Step-by-step explanation:

To find the value of tan(θ), where θ is the angle between vectors v and w, we can use the formula:

tan(θ) = sqrt((v × w)² / (v · v)(w · w))

Given that v × w = ⟨-1, 0, 4⟩ and v ⋅ w = 6, we can substitute these values into the formula to get:

tan(θ) = sqrt((-1)² + 0² + 4²) / (sqrt(v · v) * sqrt(w · w))

tan(θ) = sqrt(1 + 16) / (sqrt(v · v) * sqrt(w · w))

tan(θ) = sqrt(17) / (sqrt(v · v) * sqrt(w · w))

Since we don't have the magnitude of vectors v and w, we are unable to determine the exact value of tan(θ) without further information.