Question

Determine whether the given line is parallel to, intersects, or lies in the given plane. If the line is parallel to the plane, calculate its distance from the plane. If the line intersects the plane, find the point and angle at which they intersect.

$\mathrm{r}\left(t\right)=⟨3-2t,-4,5-4t⟩$ and $2x+5y-z=7$

Short answer

The distance between parallel to the plane is $\frac{26\sqrt{30}}{15}$

Step-by-step solution

Given information

The line $\mathrm{r}\left(t\right)=⟨3-2t,-4,5-4t⟩$ and the plane $2x+5y-z=7$

Calculation

The goal is to figure out if a given line is parallel to, intersects with, or lies in a given plane. Calculate the dot product of the distance vector and the normal vector to determine the line's behavior.

The direction vector of the line $r\left(t\right)=⟨3-2t,-4,5-4t⟩$is ${\mathbf{d}}_1=⟨-2,0,-4⟩$and the normal vector of the plane $2x+5y-z=7$is ${\mathbf{N}}_2=⟨2,5,-1⟩$

The direction vector's dot product with the normal vector is:

$$\begin{gathered} {\mathbf{d}}_1·{\mathbf{N}}_2=⟨-2,0,-4⟩·⟨2,5,-1⟩ \\ =-4+0+4 \\ =0 \end{gathered}$$

The result of the dot product is zero. As a result, the line and the plane do not cross. As a result, the lines are parallel.

The distance between two parallel planes is given by:

$\text{distance}=\frac{\left|\mathbf{N}·\overrightarrow{R_1{R}_2}\right|}{\Vert \mathbf{N}\Vert }$

Calculation

The point $R_1=(3,-4,5)$is lies on the line $r\left(t\right)=⟨t,-5-2t,-1+3t)$and the point $R_2=(0,0,-7)$lies on the plane $2x+5y-z=7$

The vector $\overrightarrow{R_1{R}_2}$is given by:

$$\begin{gathered} \overline{R_1{R}_2}=⟨0-3,0-(-4),-7-5⟩ \\ =⟨-3,4,-12⟩ \end{gathered}$$

The distance from $P$ to the plane is:

$$\begin{gathered} \text{distance}=\frac{\left|\mathbf{N}·\overrightarrow{R_1{R}_2}\right|}{\Vert \mathbf{N}\Vert } \\ =\frac{|⟨2,5,-1⟩·⟨-3,4,-12⟩|}{|⟨2,5,-1⟩|}\text{(Substitution)} \\ =\frac{|-6+20+12|}{\sqrt{4+25+1}}\text{(Dot product)} \\ =\frac{\left|26\right|}{\sqrt{30}} \\ =\frac{26\sqrt{30}}{30}\text{(Simplify)} \\ =\frac{13\sqrt{30}}{15} \end{gathered}$$

Therefore, the distance between parallel to the plane is $\frac{26\sqrt{30}}{15}$