If A = 2i + 3j and B = ai + bj, and A · B = 0, what is the relationship between

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Question: If A = 2i + 3j and B = ai + bj, and A · B = 0, what is the relationship between a and b?

Options:

Correct Answer: a = -3b/2

Solution:

A · B = 2a + 3b = 0 implies a = -3b/2.

If A = 2i + 3j and B = ai + bj, and A · B = 0, what is the relationship between a and b?

If A = 2i + 3j and B = ai + bj, and A · B = 0, what is the relationship between a and b?

Step 1: Identify the vectors A and B. A = 2i + 3j and B = ai + bj.
Step 2: Understand that the dot product A · B is calculated as (2i + 3j) · (ai + bj).
Step 3: Use the formula for the dot product: A · B = (2 * a) + (3 * b).
Step 4: Set the dot product equal to 0 because the question states A · B = 0. So, we have 2a + 3b = 0.
Step 5: Rearrange the equation 2a + 3b = 0 to find the relationship between a and b. This gives us 2a = -3b.
Step 6: Divide both sides of the equation by 2 to isolate a: a = -3b / 2.

Dot Product – The dot product of two vectors is zero if and only if the vectors are orthogonal (perpendicular) to each other.
Vector Representation – Understanding how to represent vectors in component form and manipulate their components.
Algebraic Manipulation – Solving for one variable in terms of another using algebraic techniques.