Answer:
Explanation:
To find the magnitude of the cross product of vectors a and b, we can use the formula:
|a x b| = |a| |b| sin(θ)
where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.
First, let's calculate the cross product of vectors a and b:
a x b = (2i + j + 8k) x (i + 3j + 4k)
= (j)(4k) - (8k)(3j) - (i)(4k) - (j)(2i) + (8k)(i) - (2i)(i + 3j)
= -10i - 20j - 20k
Next, let's find the magnitudes of vectors a and b:
|a| = √(2^2 + 1^2 + 8^2) = √(69)
|b| = √(1^2 + 3^2 + 4^2) = √(26)
Finally, we need to find the angle θ between vectors a and b. We can use the dot product to find cos(θ):
a · b = (2i)(i) + (1j)(3j) + (8k)(4k) = 2 + 3 + 32 = 37
|a| |b| cos(θ) = a · b
cos(θ) = a · b / (|a| |b|)
cos(θ) = 37 / (√(69) √(26))
cos(θ) ≈ 0.893
Using the fact that sin^2(θ) + cos^2(θ) = 1, we can solve for sin(θ):
sin(θ) = √(1 - cos^2(θ))
sin(θ) ≈ 0.449
Substituting the values we found into the formula for the magnitude of the cross product, we get:
|a x b| = |a| |b| sin(θ)
|a x b| = √(69) √(26) (0.449)
|a x b| ≈ 26.81
Therefore, the magnitude of the cross product of vectors a and b is approximately 26.81.