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.