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4. A line can either lie on a plane, lie parallel to it or intersect it. (6 marks) a. Determine, if there is one, the point of intersection between the line and plane given: L1: (x-7)/2 = (y-9)/4 = (z-4)/3 ??2: [x, y, z] = [5, 4, -1] + s[2, 3, 1] + t[4, -1, -2] b. Determine the angle between the line and the plane. c. Give the equation of a plane and three lines, one of which is parallel to the plane, one of which lies on the plane, and one of which intersects the plane. Explain your reasoning and include a LanGraph.

          4. A line can either lie on a plane, lie parallel to it or intersect it. (6 marks)
a. Determine, if there is one, the point of intersection between the line and plane given:

L1: (x-7)/2 = (y-9)/4 = (z-4)/3
??2: [x, y, z] = [5, 4, -1] + s[2, 3, 1] + t[4, -1, -2]

b. Determine the angle between the line and the plane.
c. Give the equation of a plane and three lines, one of which is parallel to the plane, one of which lies on the plane, and one of which intersects the plane. Explain your reasoning and include a LanGraph.

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4. A line can either lie on a plane, lie parallel to it or intersect it. (6 marks)
a. Determine, if there is one, the point of intersection between the line and plane given:

L1: (x-7)/2 = (y-9)/4 = (z-4)/3
??2: [x, y, z] = [5, 4, -1] + s[2, 3, 1] + t[4, -1, -2]

b. Determine the angle between the line and the plane.
c. Give the equation of a plane and three lines, one of which is parallel to the plane, one of which lies on the plane, and one of which intersects the plane. Explain your reasoning and include a LanGraph.

Added by Marianna R.

Calculus

Laura Taalman, Peter Kohn 1st Edition

Chapter 10

Instant Answer

Solved by Expert Sam Stansfield

04/17/2024

Step 1

Determine the point of intersection between the line and the plane **Step 1:** Write the parametric equations of the line \(L_1\). Given \(L_1: \frac{x-7}{2}=\frac{y-9}{4}=\frac{z-4}{3}\), we can write the parametric equations by setting \(\frac{x-7}{2} = Show more…

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Transcript

00:01 Okay, so we have an equation for a line, and we have the equation for a plane.

00:06 We want to know if the line intersects the plane.

00:09 So here's what i'm going to do.

00:11 I'm going to write this out as three separate equations, parametric equations for the plane.

00:24 S and t are parameters on the plane itself.

01:11 So then the equation for our line gives us really two relationships between x, y, and z.

01:20 It looks like three, but there's actually only two that are linearly independent.

01:27 So i'm going to start with the left -hand equal sign.

01:32 I'm going to plug in from up here in terms of x and y.

02:33 So now we'll take the other possible equal sign.

02:40 We had three choices.

02:41 I'm just picking another one.

02:51 So here's another equation.

03:03 So we have to solve that pair of equations.

03:05 So there's our values of s and t where the intersection occurs.

03:17 Okay, so there's the coordinates of the point of intersection between the plane and the line.

03:31 So to determine the angle between the line and the plane, we need two things.

03:37 We need the normal vector to the plane, which is pretty easy to find.

03:41 And we also want to know the direction vector of the line.

03:46 Alright? so let's focus on the line right now...

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