If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?

1 question

1 item

Q: If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?

Solution: Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.

Steps: 9

Step 1: Identify the vectors a and b. Here, a = (3, 4, 0) and b = (0, 0, 5).
Step 2: Calculate the magnitudes of vectors a and b. The magnitude of a is calculated as |a| = √(3^2 + 4^2 + 0^2).
Step 3: Calculate 3^2 = 9 and 4^2 = 16. So, |a| = √(9 + 16 + 0) = √25 = 5.
Step 4: For vector b, calculate its magnitude |b| = √(0^2 + 0^2 + 5^2).
Step 5: Calculate 5^2 = 25. So, |b| = √25 = 5.
Step 6: Find the angle between vectors a and b. Since a and b are perpendicular, the angle is 90 degrees.
Step 7: Use the formula for the magnitude of the cross product: |a × b| = |a||b|sin(θ). Here, θ = 90 degrees, so sin(90) = 1.
Step 8: Substitute the values into the formula: |a × b| = |a| * |b| * sin(90) = 5 * 5 * 1.
Step 9: Calculate the result: 5 * 5 = 25. Therefore, the magnitude of a × b is 25.