### Video Transcript

Given that vector ๐ equals nine, negative five, negative one; vector ๐ equals seven, negative ๐, negative five; vector ๐ equals 10, negative 55, ๐ minus three; and vector ๐๐ is parallel to vector ๐, find ๐ minus ๐.

In these three vectors weโre given ๐, ๐, and ๐, two of these vectors, ๐ and ๐, have unknown components. The ๐ฆ-component of vector ๐ is negative ๐ and the ๐ง-component of vector ๐ is ๐ minus three. We donโt know the values of either ๐ or ๐, but we do know that this vector ๐๐ is parallel to our vector ๐. This vector ๐๐ is the fourth vector that we donโt yet know, but we can solve for.

Graphically, if vector ๐ looked like this and starting from the same point vector ๐ looked like this, then this vector ๐๐ would go from the tip of vector ๐ to the tip of vector ๐. The way to compute the components of vector ๐๐ is to subtract the components of vector ๐ from those of vector ๐. Since weโre working with vectors, we do the subtraction component by component. For the ๐ฅ-component, we have seven minus nine or negative two; for the ๐ฆ-component, negative ๐ minus negative five or negative ๐ plus five; and for the ๐ง-component, negative five minus negative one or negative five plus one. The components of this vector ๐๐ then are negative two, negative ๐ plus five, and negative four.

Recalling that vector ๐๐ is parallel to vector ๐, we can remember that, in general, if we have two three-dimensional vectors โ weโll call them ๐ฎ and ๐ฏ โ and those vectors are parallel, then we can say this about their components. The ratio of their ๐ฅ-components equals the ratio of their ๐ฆ-components equals the ratio of their ๐ง-components. The fact that ๐๐ is parallel to vector ๐ means we can say the same thing about the components of these vectors. That is, that the ratio of the ๐ฅ-component of ๐๐ to the ๐ฅ-component of ๐ equals the ratio of the ๐ฆ-component of ๐๐ to the ๐ฆ-component of ๐ and also the ๐ง-component of ๐๐ to the ๐ง-component of ๐.

In this whole expression here, we essentially have two unknowns and two independent equations. The unknowns are ๐ and ๐. And our first independent equation is that the ๐ฅ-value ratio equals the ๐ฆ-value ratio and the second is that the ๐ฅ-value ratio equals the ๐ง-value ratio. By cross multiplying, we can solve both of these equations for ๐ and ๐. In our first equation, by multiplying both sides by negative 55 and 10, we get 110 is equal to negative 10๐ plus 50. This implies that ๐ is equal to negative six. In our second equation, multiplying both sides by ๐ minus three and 10 tells us that negative two ๐ plus six equals negative 40. That means that negative two ๐ equals negative 46 or that ๐ equals positive 23.

Knowing the values of ๐ and ๐, we can move on towards our solution. Since ๐ is negative six and ๐ is positive 23, ๐ minus ๐ equals negative 29. This is the value of the constant ๐ minus the constant ๐.