Video Transcript
Given that ๐ is the vector ๐ฅ, negative 19 and ๐ is the vector negative 19, ๐ฆ and the vector ๐ is parallel to the vector ๐, find the relationship between ๐ฅ and ๐ฆ.
In this question, weโre given two vectors, the vector ๐ and the vector ๐. And in fact, weโre told some information about vectors ๐ and vector ๐. For example, weโre told that ๐ and ๐ 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 ๐ฅ and ๐ฆ, where ๐ฅ and ๐ฆ are given as components of the vectors ๐ and ๐, respectively.
To do this, letโs 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 ๐ฎ and ๐ฏ, this is exactly the same as saying that there is some scalar constant ๐ which is not equal to zero such that ๐ฎ is equal to ๐ times ๐ฏ. In other words, we need our vectors to be a nonzero scalar multiple of each other.
So because weโre told that vector ๐ and the vector ๐ are parallel in the question, we know that ๐ is equal to ๐ times ๐ for some scalar constant ๐ not equal to zero. And weโre given the components of ๐ and ๐. So we can write this in our equation. By writing our vectors ๐ and ๐ out component-wise, we have the vector ๐ฅ, negative 19 should be equal to ๐ times the vector negative 19, ๐ฆ.
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 ๐. Multiplying every component in our vector ๐ by ๐, we get the vector ๐ฅ, negative 19 is equal to the vector negative 19๐, ๐๐ฆ. So for our vectors ๐ and ๐ to be parallel, these two vectors need to be equal.
We can use this to find the value of ๐. 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: ๐ฅ must be equal to negative 19๐ and negative 19 must be equal to ๐ times ๐ฆ.
We can rearrange both of these equations to solve for ๐. Dividing our first equation through by negative 19, we get ๐ is equal to negative ๐ฅ over 19. And dividing our second equation through by ๐ฆ, we get that ๐ should be equal to negative 19 over ๐ฆ.
And before we continue, there is one thing worth pointing out here. We know our value of ๐ฆ and our value of ๐ฅ cannot be equal to zero. If ๐ฅ was equal to zero, then ๐ would only point in the vertical direction. It would have no horizontal component, so it could not be parallel to vector ๐. Something very similar is true if ๐ฆ was equal to zero. So ๐ฅ and ๐ฆ are both not equal to zero. So we donโt need to worry about dividing through by ๐ฆ.
Now we see we have two equations for our constant ๐. Since both of these are equal to ๐, we can set them equal to each other. In other words, we know that negative ๐ฅ over 19 is equal to ๐ and negative 19 over ๐ฆ is also equal to ๐. So these two are equal. And this is in fact a relationship between ๐ฅ and ๐ฆ. However, we can simplify this even further. We could multiply through by negative ๐ฆ and we could also multiply through by 19. Doing this and simplifying, we get the equation ๐ฅ times ๐ฆ should be equal to 361, which is our final answer.
Therefore, we were able to show if the vector ๐ ๐ฅ, negative 19 and the vector ๐ negative 19, ๐ฆ are parallel, then we must have that ๐ฅ times ๐ฆ is equal to 361.
