### Video Transcript

Given that the vector ๐ is equal to negative ๐ข minus two ๐ฃ and the vector ๐ is equal to ๐๐ข minus eight ๐ฃ and the vector ๐ is parallel to the vector ๐, where ๐ข and ๐ฃ are two perpendicular unit vectors, find the value of the scalar ๐.

In this question, weโre given some information about two vectors, the vector ๐ and the vector ๐. Weโre given these vectors in terms of two perpendicular unit directional vectors, ๐ข and ๐ฃ. Weโre also told that the vector ๐ is parallel to the vector ๐. We need to use this to determine the value of the scalar ๐.

To do this, letโs start by recalling what it means for two vectors to be parallel. We say that two vectors are parallel if theyโre scalar multiples of each other. In other words, weโll say the vectors ๐ฎ and ๐ฏ are parallel if there exists some scalar ๐ such that ๐ฎ is equal to ๐ times ๐ฏ. Since weโre told that the vector ๐ is parallel to the vector ๐, there must exist some scalar ๐ such that ๐ is equal to ๐ times ๐. To help us find the value of this scalar ๐, we can substitute the expressions weโre given for vectors ๐ and ๐ in terms of the unit directional vectors ๐ข and ๐ฃ.

We get negative ๐ข minus two ๐ฃ will be equal to ๐ times ๐๐ข minus eight ๐ฃ. We can then simplify the right-hand side of this equation by remembering to multiply a vector by a scalar, we multiply all of the components of our vector by our scalar, where we remember the components of our vector will be the coefficients of the unit directional vectors ๐ข and ๐ฃ. This gives us that negative ๐ข minus two ๐ฃ will be equal to ๐๐๐ข minus eight ๐๐ฃ. Remember, for two vectors to be equal, their components must be equal. So we can compare the components of ๐ฃ on both sides of our equation. This then gives us the equation negative two must be equal to negative eight ๐, which we can solve for ๐ by dividing through by negative eight.

This then gives us that ๐ is equal to negative two divided by negative eight, which simplifies to give us one-quarter. But weโre not yet done. Remember, the question wants us to find the value of ๐. We can substitute our value of ๐ back into our equation. This then gives us the equation negative ๐ข minus two ๐ฃ will be equal to ๐ over four ๐ข minus two ๐ฃ. Remember, for both of these vectors to be equal, their components must be equal. This means we can construct an equation for ๐ by equating the components of both vectors in the ๐ข-direction.

We get that negative one must be equal to ๐ divided by four. We can then solve this equation for ๐. We multiply both sides of our equation through by four, giving us that ๐ should be equal to negative four. Therefore, we were able to show if ๐ is the vector negative ๐ข minus two ๐ฃ and ๐ is the vector ๐๐ข minus eight ๐ฃ and the vector ๐ is parallel to the vector ๐, where ๐ข and ๐ฃ are two perpendicular unit vectors, then the value of ๐ must be equal to negative four.