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

If 𝐀 = <ℎ, ℎ + 3> and 𝐁 = <3ℎ, 4ℎ − 1>, then one of the values of ℎ that makes 𝐀 ∥ 𝐁 is _.

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Video Transcript

If vector 𝐀 is equal to ℎ, ℎ plus two and vector 𝐁 is equal to three ℎ, four ℎ minus one, then one of the values of ℎ 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 𝑘 multiplied by vector 𝐁. One way of answering the question would be to substitute all four of the values in for ℎ 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 ℎ, ℎ plus two must be equal to 𝑘 multiplied by three ℎ, four ℎ 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 𝑘ℎ, 𝑘 multiplied by four ℎ minus one.

This equation will now be true when the individual components are equal. This means that ℎ must be equal to three 𝑘ℎ. 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 ℎ, giving us three 𝑘 is equal to one. Dividing both sides of this equation by three gives us a value of 𝑘 equal to one-third. If we now consider the 𝑦-components of our vector, we see that ℎ plus two is equal to 𝑘 multiplied by four ℎ minus one. Substituting 𝑘 equals one-third, the right-hand side becomes one-third multiplied by four ℎ 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 ℎ plus six. And this is equal to four ℎ minus one. We can then add one and subtract three ℎ from both sides, giving us six plus one is equal to four ℎ minus three ℎ. Simplifying this, we get a value of ℎ equal to seven. This means that the correct answer is (B). One of the values of ℎ that makes 𝐀 parallel to 𝐁 is seven.

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