### Video Transcript

Which of the following are the parametric equations of the line through point ๐ด negative eight, eight with a direction vector perpendicular to ๐ฎ is equal to the vector negative six, seven? Is it option (A) ๐ฅ is equal to negative eight minus six ๐, and ๐ฆ is equal to eight plus seven ๐? Option (B) ๐ฅ is equal to negative eight plus eight ๐, and ๐ฆ is equal to negative six plus seven ๐. Is it option (C) ๐ฅ is equal to negative eight plus seven ๐, and ๐ฆ is equal to eight minus six ๐? Or is it option (D) ๐ฅ is equal to negative eight plus seven ๐, and ๐ฆ is equal to eight plus six ๐?

In this question, weโre asked to determine which of four given options are the parametric equations of a line passing through a given point perpendicular to a given vector. And to do this, letโs start by recalling what we mean by the parametric equations of a line. Theyโre two equations of the form ๐ฅ is equal to ๐ฅ sub zero plus ๐ times ๐ and ๐ฆ is equal to ๐ฆ sub zero plus ๐ multiplied by ๐. ๐ is called our scalar, and it can take any value. And we know since ๐ can be equal to zero, we can substitute this into the equation to note that ๐ฅ sub zero, ๐ฆ sub zero is a point which lies on the line. And in fact, we can choose any point which lies on the line for the point ๐ฅ sub zero, ๐ฆ sub zero; this will give us different equations for the line.

Similarly, we can recall that vector ๐, ๐ is a nonzero vector which is parallel to the line. Therefore, to find the parametric equations of a line, we need a point which lies on the line. And we also need a nonzero vector parallel to the line. And weโre given in the question that the line passes through the point with coordinates negative eight, eight. So, we can choose ๐ฅ sub zero to be equal to negative eight and ๐ฆ sub zero to be equal to eight. This is one possible choice of a point which lies on the line.

And if we look at our four given options, we can see three of these given options have chosen the same point. The value of ๐ฅ sub zero in these equations is negative eight and the value of ๐ฆ sub zero is eight. However, in equation (B), we can notice something interesting. The value of ๐ฆ sub zero is negative six. Itโs possible to use this information to eliminate this option since we know the line is not vertical. And for this line to pass through both ๐ด, which has coordinates negative eight, eight, and this point ๐ฅ sub zero, ๐ฆ sub zero, which is the point negative eight, negative six, the line would need to be vertical and then it wouldnโt be perpendicular to our vector ๐ฎ. However, to fully justify this, letโs find a direction vector of this line.

We want to do this by using the fact that our line is perpendicular to the vector ๐ฎ. And there are many different ways we can use this information to determine a direction vector of the line. For example, for any two-dimensional vector ๐ฎ, we can determine a vector perpendicular to this vector by switching its components and then multiplying one of the nonzero components by negative one. Switching the components of vector ๐ฎ gives us the vector seven, negative six. And then if we switch the sign of the vertical component of this vector, we get the vector seven, six. We can call this vector ๐. And for due diligence, letโs check that this vector is in fact perpendicular to ๐ฎ.

We can do this by recalling two vectors are perpendicular when their dot product is equal to zero. So, we want to find the dot product between vectors ๐ฎ and ~~๐ฏ~~ [๐]. We need to find the dot product of the vector negative six, seven and the vector seven, six. And we do this by finding the sum of the products of the corresponding components. We get negative six times seven plus seven times six, which we can evaluate is equal to zero. Therefore, weโve confirmed that these two vectors are perpendicular, and this means that ๐ is a direction vector of our line. In fact, any nonzero scalar multiple of ๐ will be a direction vector of this line.

So, one possible choice of our direction vector ๐, ๐ is to set ๐ equal to seven and ๐ equal to six. And we can see that this is exactly what is written in option (D). The coefficient of ๐ in our ๐ฅ- equation is seven and the coefficient of ๐ in the ๐ฆ-equation is six. And for due diligence, itโs worth noting we can use this fact to eliminate all three of the other options. For example, the vector seven, negative six is not parallel to vector ๐, and itโs also not perpendicular to vector ๐ฎ. We can show this in many different ways. For example, we could show itโs not a scalar multiple of ๐. Or we could calculate its dot product with ๐ฎ, which would be nonzero. And we can also follow the same process to eliminate options (A) and (B).

Therefore, by substituting ๐ฅ sub zero is equal to negative eight, ๐ฆ sub zero is equal to eight, ๐ is equal to seven, and ๐ is equal to six into our parametric equations, we were able to show that ๐ฅ is equal to negative eight plus seven ๐ and ๐ฆ is equal to eight plus six ๐ is the parametric equations of the line through point ๐ด is negative eight, eight with a direction vector perpendicular to ๐ฎ is the vector negative six, seven. This is option (D).