Given that vector 𝐀 equals nine, negative five, negative one; vector 𝐁 equals seven, negative 𝑘, negative five; vector 𝐂 equals 10, negative 55, 𝑚 minus three; and vector 𝐀𝐁 is parallel to vector 𝐂, find 𝑘 minus 𝑚.
In these three vectors we’re given 𝐀, 𝐁, and 𝐂, two of these vectors, 𝐁 and 𝐂, have unknown components. The 𝑦-component of vector 𝐁 is negative 𝑘 and the 𝑧-component of vector 𝐂 is 𝑚 minus three. We don’t know the values of either 𝑘 or 𝑚, but we do know that this vector 𝐀𝐁 is parallel to our vector 𝐂. This vector 𝐀𝐁 is the fourth vector that we don’t yet know, but we can solve for.
Graphically, if vector 𝐀 looked like this and starting from the same point vector 𝐁 looked like this, then this vector 𝐀𝐁 would go from the tip of vector 𝐀 to the tip of vector 𝐁. The way to compute the components of vector 𝐀𝐁 is to subtract the components of vector 𝐀 from those of vector 𝐁. Since we’re working with vectors, we do the subtraction component by component. For the 𝑥-component, we have seven minus nine or negative two; for the 𝑦-component, negative 𝑘 minus negative five or negative 𝑘 plus five; and for the 𝑧-component, negative five minus negative one or negative five plus one. The components of this vector 𝐀𝐁 then are negative two, negative 𝑘 plus five, and negative four.
Recalling that vector 𝐀𝐁 is parallel to vector 𝐂, we can remember that, in general, if we have two three-dimensional vectors — we’ll call them 𝐮 and 𝐯 — and those vectors are parallel, then we can say this about their components. The ratio of their 𝑥-components equals the ratio of their 𝑦-components equals the ratio of their 𝑧-components. The fact that 𝐀𝐁 is parallel to vector 𝐂 means we can say the same thing about the components of these vectors. That is, that the ratio of the 𝑥-component of 𝐀𝐁 to the 𝑥-component of 𝐂 equals the ratio of the 𝑦-component of 𝐀𝐁 to the 𝑦-component of 𝐂 and also the 𝑧-component of 𝐀𝐁 to the 𝑧-component of 𝐂.
In this whole expression here, we essentially have two unknowns and two independent equations. The unknowns are 𝑘 and 𝑚. And our first independent equation is that the 𝑥-value ratio equals the 𝑦-value ratio and the second is that the 𝑥-value ratio equals the 𝑧-value ratio. By cross multiplying, we can solve both of these equations for 𝑘 and 𝑚. In our first equation, by multiplying both sides by negative 55 and 10, we get 110 is equal to negative 10𝑘 plus 50. This implies that 𝑘 is equal to negative six. In our second equation, multiplying both sides by 𝑚 minus three and 10 tells us that negative two 𝑚 plus six equals negative 40. That means that negative two 𝑚 equals negative 46 or that 𝑚 equals positive 23.
Knowing the values of 𝑘 and 𝑚, we can move on towards our solution. Since 𝑘 is negative six and 𝑚 is positive 23, 𝑘 minus 𝑚 equals negative 29. This is the value of the constant 𝑘 minus the constant 𝑚.