Find parametric equations for the line passing through the...

Find parametric equations for the line passing through the point P(1, -4, 2) and perpendicular to the plane $4x + 2y - z = 6$. 1. $x = 4 + t$, $y = 2 + 4t$, $z = 1 - 2t$ 2. $x = 1 + 4t$, $y = -4 + 2t$, $z = 2 - t$ 3. $x = 4 - t$, $y = -2 + 4t$, $z = -1 + 2t$ 4. $x = 4 + t$, $y = 2 - 4t$, $z = -1 + 2t$ 5. $x = 1 - 4t$, $y = 4 - 2t$, $z = 2 - t$ 6. $x = -1 + 4t$, $y = 4 + 2t$, $z = -2 - t$

Transcript

00:01 So in this question, we say we want to find a parametric equation for the line that passes through the point 214, that is perpendicular to the plane 3x plus y plus z equals 3.

00:20 So let's figure out the normal vector to this plane.

00:26 So the normal vector to this plane is going to be the vector that has, as its components, the coefficients of x, y, and z.

00:39 So 3, 1, and 1, that is my normal vector to that plane.

00:46 And so if we are perpendicular to that plane, we are in the direction of that normal vector to the plane.

00:55 And so now i can write my parametric equations for this line.

01:01 My x, y, and z, each star, at 2, 1, and four respectively.

01:10 And to that, i add my x, y, and z components of the normal vector.

01:16 So my x is equal to 2 plus 3t, my y is equal to 1 plus t, and my z is equal to 4 plus t.

01:26 And that would be my answer to this first question.

01:31 In the second question, we want to find an equation of the plane that contains the line, in r of t, which has components t, 2t, and 3t, as well as the points negative 1, 4, and 1.

01:47 So in order to write the equation of this plane, i'm going to need two vectors that lie in this plane.

01:55 So first of all, consider r0.

01:59 R of 0 would be 0 -0, and so 0 -0 is a point that lies in this plane.

02:08 And so one vector that is going to lie in this plane is going to extend from the point negative 1 -4 -1 to the point 0 -0.

02:24 Such a vector does what? well, if it's going from negative 1 to 0, we're going up 1 in the x direction.

02:33 I'm going down 4 in the y direction.

02:36 If i start at y equals 4 and end at y equals 0, and negative 1 in the z direction, if i start at z equals 1 and i end at z equals 0...

Question

Please give Ace some feedback