Question

Let P be the plane given by the equation 2x + y − 3z = 2. The point Q(1, 2, 3) is not on the plane P, the point R is on the plane P, and the line L1 through Q and R is orthogonal to the plane P. Let W be another point (1, 1, 3). Find parametric equations of the line L2 that passes through points W and R.

Answer #1

1. Solve all three:
a. Determine whether the plane 2x + y + 3z – 6 = 0 passes
through the points (3,6,-2) and (-1,5,-1)
b. Find the equation of the plane that passes through the points
(2,2,1) and (-1,1,-1) and is perpendicular to the plane 2x - 3y + z
= 3.
c. Determine whether the planes are parallel, orthogonal, or
neither. If they are neither parallel nor orthogonal, find the
angle of intersection:
3x + y - 4z...

(a) Find the parametric equation of the straight line with
non-parametric equation 2x ? y = 3.
(b) Find the parametric equation of L1 if L1 is a straight line
in 3D passing through the points (1, 2, 3) and (4, 4, ?2). (c) Show
that L1 also passes through the point (?5, ?2, 13).

given the plane P:2y-3z-6=0, find scalar parametric equations for
the line perpendicular to P that passes through the point
(-1,-2,3)

2. Let P 1 and P2 be planes with general equations P1 : −2x + y
− 4z = 2, P2 : x + 2y = 7.
(a) Let P3 be a plane which is orthogonal to both P1 and P2. If
such a plane P3 exists, give a possible general equation for it.
Otherwise, explain why it is not possible to find such a plane. (b)
Let ` be a line which is orthogonal to both P1 and P2....

(a) Find the distance between the skew lines l1 and l2 given
with the vector equations l1 : r1(t) = (1+t)i+ (1+6t)j+ (2t)k; l2 :
r2(s) = (1+2s)i+ (5+15s)j+ (−2+6s)k.
(b) Determine if the plane given by the Cartesian equation −x +
2z = 0 and the line given by the parametric equations x = 5 + 8t, y
= 2 − t, z = 10 + 4t are orthogonal, parallel, or neither.

Find an equation of the plane. The plane that passes through the
point (−3, 3, 2) and contains the line of intersection of the
planes x + y − z = 2 and 2x − y + 4z = 1

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

Find an equation of the plane.
The plane that passes through the line of intersection of the
planes
x − z = 2 and y + 3z = 1
and is perpendicular to the plane
x + y − 3z = 3

1/ Find linear equation for the plane containing (-1,2,1) that
is parallel to the plane 2x - y + 3z = 1
2/ Find linear equation for the plane containing (2,0,9) that is
perpendicular to the line (x-2)/5 = (y+4)/3 = z/2

let S be the surface defined by x^4-2x^2y^2+3z^2=12, Find the
equation of the tangent plane to the surface S at (0,1,2).

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