If A = 4i + 3j and B = ai + bj, and A · B = 24, find the value of a + b if a = 2

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Question: If A = 4i + 3j and B = ai + bj, and A · B = 24, find the value of a + b if a = 2.

Options:

Correct Answer: 12

Solution:

A · B = (4)(2) + (3)(b) = 8 + 3b = 24, 3b = 16, b = 16/3. a + b = 2 + 16/3 = 10.

If A = 4i + 3j and B = ai + bj, and A · B = 24, find the value of a + b if a = 2.

If A = 4i + 3j and B = ai + bj, and A · B = 24, find the value of a + b if a = 2.

Step 1: Identify the vectors A and B. A = 4i + 3j and B = ai + bj.
Step 2: Substitute the value of a into vector B. Since a = 2, B becomes 2i + bj.
Step 3: Calculate the dot product A · B. The formula for the dot product is A · B = (4)(2) + (3)(b).
Step 4: Simplify the dot product calculation. This gives us A · B = 8 + 3b.
Step 5: Set the dot product equal to 24, as given in the question. So, 8 + 3b = 24.
Step 6: Solve for b. Subtract 8 from both sides: 3b = 24 - 8, which simplifies to 3b = 16.
Step 7: Divide both sides by 3 to find b. So, b = 16 / 3.
Step 8: Now, find the value of a + b. Since a = 2 and b = 16/3, we calculate a + b = 2 + 16/3.
Step 9: Convert 2 into a fraction with a denominator of 3. This gives us 2 = 6/3.
Step 10: Add the fractions: 6/3 + 16/3 = (6 + 16) / 3 = 22 / 3.
Step 11: The final answer for a + b is 22 / 3.

Dot Product – Understanding the dot product of two vectors and how to calculate it using their components.
Algebraic Manipulation – Solving equations for unknown variables and performing arithmetic operations.