91影视

A point $P(0,1,-2)$ to the line by $r\left(t\right)=(1+5t,-6+t,-4t)$

The goal is to calculate the distance between the point and the line.

The formula for the distance is $\frac{||d脳{\stackrel{鈫�}{P}}_{0}P||}{鈥�d鈥�}$

Let the point $P$is $P(0,1,-2)$

The point $P_0$ on the line equation $r\left(t\right)=(1+5t,-6+t,-4t)$ is $(1,-6,0)$

The direction vector $\stackrel{鈫�}{P_{0}P}=(0-1,1-(-6),-2-0)$

Then $\stackrel{鈫�}{P_{0}P}=(-1,7,-2)$

Take the direction vector of the equation $r\left(t\right)=(1+5t,-6+t,-4t)$is $d=(5,1,-4)$

Substitute the values $d=(5,1,-4)$and $\stackrel{鈫�}{P_{0}P}=(-1,7,-2)$in the formula $\underset{炉}{鈥�d脳\stackrel{鈫�}{P_{0}P}}$

Then the distance $=\frac{鈥�(5,1,-4)脳(-1,7,-2)鈥�}{鈥�(5,1,-4)鈥�}$

The cross product of $(5,1,-4)脳(-1,7,-2)$ is calculated as follows,

$$\begin{gathered} =i(-2+28)-j(-10-4)+k(35+1) \\ =26i+14j+36k \end{gathered}$$

Thus,

On further simplification,

Distance $=\sqrt{\frac{1084}{21}}$ units

Thus, the distance between the point $P(0,1,-2)$ and the line $r\left(t\right)=(1+5t,-6+t,-4t)$ is $\sqrt{\frac{1084}{21}}$ units.

Therefore, the answer is $\sqrt{\frac{1084}{21}}$ units.