(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Ex: Find the equation of the plane passing through A(1, 0, 1), B(0, 1, 2) and C(1, 3, 2).

Solution: In this case, we first need to establish a normal vector for the plane. Note that

AB and AC are both vectors on (or parallel to) the plane. Since their cross product is

perpendicular to both AB and AC, it must be a normal vector to the plane as well. Thus,

we have n = AB × AC = [−1, 1, 1] × [0, 3, 1] = [−2, 1,−3]. Hence, the general equation

takes the form −2x + y − 3z = d, where d can be found by substituting the point A, i.e.

d = −2(1) + (0) − 3(1) = −5. Hence, the equation of the plane is −2x + y − 3z = −5.

2. Relevant equations

3. The attempt at a solution

I'm having a hard time understanding the answer - I was hoping someone could clarify a few questions that I had. It feels like im missing gigantic gaps in my vector mathematics knowledge.

Note that AB and AC are both vectors on (or parallel to) the plane.

How do I know this?

I'd normally find out if vectors are parallel if I use the dot product and end up with a 0 or 180 degree angle.

Since their cross product is perpendicular to both AB and AC, it must be a normal vector to the plane as well.

Using the cross product method, I'm left with the vector (-2, 1, 3). Using the dot product method on bot AB and AC, i find that they are equal to zero, therefore perpendicular. But, why is this normal to the plane?

I understand the rest of the solution though, but I'm majorly stuck with the above parts.

Is there an easier way to solve this? I've had to do around 8 calculations to solve this question without making major assumptions.

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# Homework Help: Equation of a plane

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