If A = 5i + 2j + 3k and B = 4i - j + 2k, find the scalar product A · B.

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Question: If A = 5i + 2j + 3k and B = 4i - j + 2k, find the scalar product A · B.

Options:

Correct Answer: 28

Solution:

A · B = (5)(4) + (2)(-1) + (3)(2) = 20 - 2 + 6 = 24.

If A = 5i + 2j + 3k and B = 4i - j + 2k, find the scalar product A · B.

If A = 5i + 2j + 3k and B = 4i - j + 2k, find the scalar product A · B.

Step 1: Identify the components of vector A. A = 5i + 2j + 3k means A has components: A_x = 5, A_y = 2, A_z = 3.
Step 2: Identify the components of vector B. B = 4i - j + 2k means B has components: B_x = 4, B_y = -1, B_z = 2.
Step 3: Use the formula for the scalar product (dot product) A · B = A_x * B_x + A_y * B_y + A_z * B_z.
Step 4: Substitute the values into the formula: A · B = (5)(4) + (2)(-1) + (3)(2).
Step 5: Calculate each term: (5)(4) = 20, (2)(-1) = -2, (3)(2) = 6.
Step 6: Add the results together: 20 - 2 + 6 = 24.

Vector Operations – Understanding how to compute the scalar (dot) product of two vectors using their components.
Component Multiplication – Applying the formula for the dot product, which involves multiplying corresponding components of the vectors and summing the results.