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

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

Options:

Correct Answer: 12

Solution:

A · B = (2)(1) + (3)(-2) + (1)(4) = 2 - 6 + 4 = 0.

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

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

Step 1: Identify the components of vector A. A = 2i + 3j + k means A has components: A_x = 2, A_y = 3, A_z = 1.
Step 2: Identify the components of vector B. B = i - 2j + 4k means B has components: B_x = 1, B_y = -2, B_z = 4.
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 = (2)(1) + (3)(-2) + (1)(4).
Step 5: Calculate each term: (2)(1) = 2, (3)(-2) = -6, (1)(4) = 4.
Step 6: Add the results together: 2 - 6 + 4.
Step 7: Simplify the expression: 2 - 6 = -4, then -4 + 4 = 0.
Step 8: Conclude that the scalar product A · B = 0.

Vector Operations – Understanding how to perform operations on vectors, specifically the scalar (dot) product.
Component Multiplication – Knowledge of multiplying corresponding components of vectors and summing the results.
Vector Notation – Familiarity with vector notation and the representation of vectors in terms of their components.