If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product

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Question: If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product A × B?

Options:

Correct Answer: √14

Solution:

Magnitude |A × B| = √(1^2 + 2^2 + 3^2) = √14.

If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product A × B?

If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product A × B?

Correct Answer: √14

Step 1: Identify the components of vector A, which are (1, 2, 3).
Step 2: Identify the components of vector B, which are (4, 5, 6).
Step 3: Calculate the cross product A × B using the formula: A × B = (A2*B3 - A3*B2, A3*B1 - A1*B3, A1*B2 - A2*B1).
Step 4: Substitute the values: A × B = (2*6 - 3*5, 3*4 - 1*6, 1*5 - 2*4).
Step 5: Calculate each component: A × B = (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3).
Step 6: Find the magnitude of the vector A × B using the formula: |A × B| = √((-3)^2 + 6^2 + (-3)^2).
Step 7: Calculate the squares: |A × B| = √(9 + 36 + 9).
Step 8: Add the squares: |A × B| = √54.
Step 9: Simplify √54 to √(9*6) = 3√6.

Vector Product – The vector product (or cross product) of two vectors results in a vector that is perpendicular to both original vectors, and its magnitude is given by |A × B| = |A||B|sin(θ), where θ is the angle between the vectors.
Magnitude Calculation – The magnitude of a vector is calculated using the formula √(x^2 + y^2 + z^2) for a vector represented as (x, y, z).