If I = (a, b, c) and J = (2, 2, 2) such that I · J = 12, what is the relationship between a, b, c?
Correct Answer: a + b + c = 6
- Step 1: Understand that I = (a, b, c) is a vector with components a, b, and c.
- Step 2: Understand that J = (2, 2, 2) is another vector with all components equal to 2.
- Step 3: The dot product I · J means we multiply corresponding components of I and J and then add them together.
- Step 4: Calculate the dot product: I · J = a * 2 + b * 2 + c * 2.
- Step 5: This simplifies to 2a + 2b + 2c.
- Step 6: We know from the question that I · J = 12, so we set up the equation: 2a + 2b + 2c = 12.
- Step 7: To simplify the equation, divide everything by 2: a + b + c = 6.
- Step 8: The relationship between a, b, and c is that their sum equals 6.
- Dot Product – Understanding the dot product of two vectors and how to derive relationships between their components.
- Linear Equations – Formulating and solving linear equations based on the results of the dot product.
