If a normal vector of the plane α is n=(4,1,1) and a direction vector of the straight line l is α=(-2,-3,3), then the sine value of the angle between l and α is

If a normal vector of the plane α is n=(4,1,1) and a direction vector of the straight line l is α=(-2,-3,3), then the sine value of the angle between l and α is

Let the angle between the direction vector of a line and the normal vector of a plane be W
N.α=4*(-2)+1*(-3)+1*3=-8
| N |=√(16+1+1)=3√2
|α|=√(4+9+9)=√22
Then cosW=n.α/|n =-8/(3√2 22)=-4/3√11=-4√11/33
The sine of the angle between l and α is 4√11/33

If the direction vector a of the line l =(-2,3,1) a normal vector n of the plane z =(4,0,1), then the cosine of the angle formed by the line l and the plane z is? Whole process

Let the angle between the line l and the plane z be θ
Sinθ=│(a·n/│a││n│)│
=│-7/√14×√17│
=√34/34
∵θ∈[0,π/2]
Cos θ=√1022/34

Let the angle between the line l and the plane z be θ
Sinθ=│(a·n/│a│││n│)│
=│-7/√14×√17│
=√34/34
∵θ∈[0,π/2]
Cos θ=√1022/34

Can a plane be proved to be parallel to a straight line if the normal vector of the a plane is perpendicular to the direction vector of the the line?

Row, because the direction vectors of two lines are perpendicular, then one line is parallel to (or on) all planes passing through the other line

How do I distinguish between Line-Plane Parallel and Line-On-Plane? The direction vector s is multiplied by the normal vector n, and then?

Usually, parallel lines are not in plane, and the product of direction vector and normal vector is perpendicular.

Usually, parallel lines are not in the plane, and the product of direction vector and normal vector is perpendicular.

If a normal vector n=(3,3,0) of the plane α and a direction vector a=(1,1,1) of the straight line L, then the cosine of the angle between L and the plane α is?

If a normal vector n of the plane α=(3,3,0) and a direction vector a of the straight line L =(1,1,1), then the cosine value of the angle between L and the plane α is √3/3

Explain why the sine value of a line and a plane is equal to the cosine value of a normal line and a normal vector of the plane

This line is not a line perpendicular to the line, this term seems a bit informal.
If it is the normal vector of a straight line, it seems that the two angles are equal.