Video: Solving Problems Involving Parallel and Perpendicular Vectors in 2D

If 𝐀 = <β„Ž, β„Ž + 3> and 𝐁 = <3β„Ž, 4β„Ž βˆ’ 1>, then one of the values of β„Ž that makes 𝐀 βˆ₯ 𝐁 is οΌΏ.

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Video Transcript

If vector 𝐀 is equal to β„Ž, β„Ž plus two and vector 𝐁 is equal to three β„Ž, four β„Ž minus one, then one of the values of β„Ž that makes vector 𝐀 parallel to vector 𝐁 is what. Is it (A) five, (B) seven, (C) negative five, or (D) negative seven?

We recall that two vectors 𝐀 and 𝐁 are parallel if vector 𝐀 is equal to some nonzero constant π‘˜ multiplied by vector 𝐁. One way of answering the question would be to substitute all four of the values in for β„Ž and see which one satisfies the property. Alternatively, we can solve the question using an algebraic method. For the two vectors to be parallel, the vector β„Ž, β„Ž plus two must be equal to π‘˜ multiplied by three β„Ž, four β„Ž minus one. We can multiply any vector by a scalar by multiplying each of the individual components by that scalar. The right-hand side simplifies to three π‘˜β„Ž, π‘˜ multiplied by four β„Ž minus one.

This equation will now be true when the individual components are equal. This means that β„Ž must be equal to three π‘˜β„Ž. As none of the options given are zero, we are looking for a solution that is not equal to zero. This means that we can divide through by β„Ž, giving us three π‘˜ is equal to one. Dividing both sides of this equation by three gives us a value of π‘˜ equal to one-third. If we now consider the 𝑦-components of our vector, we see that β„Ž plus two is equal to π‘˜ multiplied by four β„Ž minus one. Substituting π‘˜ equals one-third, the right-hand side becomes one-third multiplied by four β„Ž minus one.

There are many ways to solve this equation. One way is to multiply both sides by three. The left-hand side becomes three β„Ž plus six. And this is equal to four β„Ž minus one. We can then add one and subtract three β„Ž from both sides, giving us six plus one is equal to four β„Ž minus three β„Ž. Simplifying this, we get a value of β„Ž equal to seven. This means that the correct answer is (B). One of the values of β„Ž that makes 𝐀 parallel to 𝐁 is seven.

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