The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
Practice Questions
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Q1
The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
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A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.
Questions & Step-by-step Solutions
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Q
Q: The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
Solution: A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.
Steps: 10
Step 1: Identify the vectors A and B. A = (a, b, c) and B = (1, 2, 3).
Step 2: Substitute the value of a into vector A. Since a = 2, A becomes (2, b, c).
Step 3: Write the formula for the scalar product (dot product) of vectors A and B. The formula is A · B = 2*1 + b*2 + c*3.
Step 4: Calculate the scalar product using the values. This gives us 2*1 + b*2 + c*3 = 2 + 2b + 3c.
Step 5: Set the scalar product equal to 14, as given in the question. So, we have 2 + 2b + 3c = 14.
Step 6: Simplify the equation. Subtract 2 from both sides: 2b + 3c = 14 - 2, which simplifies to 2b + 3c = 12.
Step 7: We need to find the value of b + c. To do this, we can express one variable in terms of the other. Let's isolate b: 2b = 12 - 3c, so b = (12 - 3c) / 2.
Step 8: Now, we can express b + c in terms of c. Substitute b into b + c: b + c = (12 - 3c) / 2 + c.
Step 9: Combine the terms: b + c = (12 - 3c + 2c) / 2 = (12 - c) / 2.
Step 10: To find a specific value, we can set c = 0, which gives us b + c = 12 / 2 = 6.
