Our professor has tasked us with solving these 4 questions (not-for-marks homework) WITHOUT using the cross product. I can only solve them with the cross product:
Here are the answers we're given:
Can anyone explain the process to solving at least 1 of these? I believe starting requires finding the normal vector.
Best Answer
Consider the first problem. Let's write the vector $\vec{x} = (x, y, z)$ in terms of its coordinates, and let's use different letters just to improve clarity. The vector equation is simply an equivalent formulation of the follow system of linear equations: $$ \begin{align} x &= 1 + 2s + 4t & (1)\\ y &= 4 + 3s + t & (2) \\ z &= 7 - s &(3) \end{align} $$ The goal is to eliminate the auxiliary variables $s$ and $t$ so that we get a single equation in $x, y, z$. For example, we can use $(1)$ and $(2)$ to eliminate $t$ first, so that $(1) - 4 \times (2)$ gives us $x - 4y = -15 -10s$, which we can call equation $(4)$. By combining $(4)$ and $(3)$ suitably, we can eliminate $s$.