Q.Find the vector equation of the plane passing through the point having position vector $\hat{i} + \hat{j} + \hat{k}$ and perpendicular to the vector $4\hat{i} + 5\hat{j} + 6\hat{k}$.

[!TLDR] For a plane through point $A(\vec a)$ perpendicular to $\vec n$, the equation is $\vec r\cdot\vec n = \vec a\cdot\vec n$; just compute the one …

Here the plane passes through the point with position vector $\vec a = \hat i+\hat j+\hat k$ and is perpendicular to $\vec n = 4\hat i+5\hat j+6\hat k$.

By Theorem 6.8, the equation of the plane is $\vec r\cdot\vec n = \vec a\cdot\vec n$.

Compute $\vec a\cdot\vec n$: …

1. Identify the point's position vector $\vec a$ and the normal $\vec n$ from the question.
2. Compute the dot product $\vec a\cdot\vec n$. …

Confusing which vector is the point and which is the normal when both are similar-looking sums of $\hat i,\hat j,\hat k$; writing the equation as $\vec r\cdot\vec a = \vec n \cdot \vec n$ or …