Given that 𝐀 is the vector 𝑥, negative 19 and 𝐁 is the vector negative 19, 𝑦 and the vector 𝐀 is parallel to the vector 𝐁, find the relationship between 𝑥 and 𝑦.
In this question, we’re given two vectors, the vector 𝐀 and the vector 𝐁. And in fact, we’re told some information about vectors 𝐀 and vector 𝐁. For example, we’re told that 𝐀 and 𝐁 are parallel. This is represented by the two vertical lines between our vectors in the question. We need to use all of this information to determine the relationship between 𝑥 and 𝑦, where 𝑥 and 𝑦 are given as components of the vectors 𝐀 and 𝐁, respectively.
To do this, let’s start by recalling what we mean when we say that two vectors are parallel. We say that two vectors are parallel if they point in the same direction or in exactly opposite directions. For vectors 𝐮 and 𝐯, this is exactly the same as saying that there is some scalar constant 𝑘 which is not equal to zero such that 𝐮 is equal to 𝑘 times 𝐯. In other words, we need our vectors to be a nonzero scalar multiple of each other.
So because we’re told that vector 𝐀 and the vector 𝐁 are parallel in the question, we know that 𝐀 is equal to 𝑘 times 𝐁 for some scalar constant 𝑘 not equal to zero. And we’re given the components of 𝐀 and 𝐁. So we can write this in our equation. By writing our vectors 𝐀 and 𝐁 out component-wise, we have the vector 𝑥, negative 19 should be equal to 𝑘 times the vector negative 19, 𝑦.
Now remember, when we multiply a vector by a scalar, we do this component-wise. In other words, we need to multiply every single component of our vector by 𝑘. Multiplying every component in our vector 𝐁 by 𝑘, we get the vector 𝑥, negative 19 is equal to the vector negative 19𝑘, 𝑘𝑦. So for our vectors 𝐀 and 𝐁 to be parallel, these two vectors need to be equal.
We can use this to find the value of 𝑘. Remember, for two vectors to be equal, they must have the same number of components and all of their components have to be equal. So for these two vectors to be equal, their horizontal components must be equal and their vertical components must be equal. Setting these to be equal, we get two equations which must be true: 𝑥 must be equal to negative 19𝑘 and negative 19 must be equal to 𝑘 times 𝑦.
We can rearrange both of these equations to solve for 𝑘. Dividing our first equation through by negative 19, we get 𝑘 is equal to negative 𝑥 over 19. And dividing our second equation through by 𝑦, we get that 𝑘 should be equal to negative 19 over 𝑦.
And before we continue, there is one thing worth pointing out here. We know our value of 𝑦 and our value of 𝑥 cannot be equal to zero. If 𝑥 was equal to zero, then 𝐀 would only point in the vertical direction. It would have no horizontal component, so it could not be parallel to vector 𝐁. Something very similar is true if 𝑦 was equal to zero. So 𝑥 and 𝑦 are both not equal to zero. So we don’t need to worry about dividing through by 𝑦.
Now we see we have two equations for our constant 𝑘. Since both of these are equal to 𝑘, we can set them equal to each other. In other words, we know that negative 𝑥 over 19 is equal to 𝑘 and negative 19 over 𝑦 is also equal to 𝑘. So these two are equal. And this is in fact a relationship between 𝑥 and 𝑦. However, we can simplify this even further. We could multiply through by negative 𝑦 and we could also multiply through by 19. Doing this and simplifying, we get the equation 𝑥 times 𝑦 should be equal to 361, which is our final answer.
Therefore, we were able to show if the vector 𝐀 𝑥, negative 19 and the vector 𝐁 negative 19, 𝑦 are parallel, then we must have that 𝑥 times 𝑦 is equal to 361.