Determining the distances between points and planes

[ttpp1124]

Homework Statement

Hi! I finished the question, can someone check? Also, should I be reducing my answer?

Relevant Equations

n/a

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[Mark44]

Homework Statement:: Hi! I finished the question, can someone check? Also, should I be reducing my answer?
Relevant Equations:: n/a

https://www.physicsforums.com/attachments/260124

Looks fine to me. The formula you used apparently was
$d = |\vec v|\sin(\theta)$, where $\vec v$ is the vector from a point on the line Q(6, 0, 1) and P(1, -5, 2), and $\theta$ is the angle between $\overrightarrow{QP}$ and the direction vector for the line, <3, 1, 2>.
The magnitude of the cross product of $\vec u$ and $\vec v$, is $|\vec u \times \vec v| =|\vec u||\vec v|\sin(\theta)$. With a bit of algebra, you get $d = \frac{|\vec u \times \vec v|}{|\vec u|}$.
I don't see that your final fraction will simplify very much. The only common factor of 390 and 14 is a single factor of 2.

I also don't see any simple way of checking, other than using the dot product to get the projection of $\overrightarrow{PQ}$ onto the line's direction vector (which will be the base of the right triangle), and then using Pythogoras to get the height of the triangle (which will be the distance from the line to the point.

Something I think would be useful would be to draw a sketch of a line and the given point, and derive either of the formulas for distance. This is more useful, IMO, than merely memorizing a canned formula.