Mathematics

Mathematics

Problem

Find the equation of the plane passing through (3,4,−1)(3, 4, -1), which is parallel to the plane r⋅(2i^−3j^+5k^)+7=0\mathbf{r} \cdot (2\hat{i}-3\hat{j}+5\hat{k})+7=0 .
Intuitive explanation

Parallel planes share the same normal vector, so your new plane has the same normal n=2i^−3j^+5k^\mathbf{n} = 2\hat{i}-3\hat{j}+5\hat{k}. The key is using the point-normal form: plug in the given point and normal to find the constant term that makes your plane pass through (3,4,−1)(3, 4, -1). This shifts the plane without changing its orientation.

Intuitive explanation

Parallel planes share the same normal vector, so your new plane has the same normal n=2i^−3j^+5k^\mathbf{n} = 2\hat{i}-3\hat{j}+5\hat{k}. The key is using the point-normal form: plug in the given point and normal to find the constant term that makes your plane pass through (3,4,−1)(3, 4, -1). This shifts the plane without changing its orientation.

Browse the problem library