Question

The planes x + 3 y − 2 z = 1 and 2 x − y + 3 z = 4 intersect in a line. A direction vector for this line is given by:

Answer #1

1. Determine whether the lines are parallel, perpendicular or
neither. (x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 =
(z-2)/6
2. A) Find the line intersection of vector planes given by the
equations -2x+3y-z+4=0 and 3x-2y+z=-2
B) Given U = <2, -3, 4> and V= <-1, 3, -2> Find a. U
. V b. U x V

Complete the parametric equations for the line where the planes
− 2 y − 12 z = − 8 and − 6 x + 7 y − 3 z = − 17 intersect. x ( t )
= 90 t. Find y ( t ) and z ( t ).

given the planes P1 and P2 determine whether the planes intersect
or are parallel. if they intersect, give the direction vector for
thr line of intersection. P1:3x+y-3z=4
p2:2x+3y+2z=6

The planes z+5?+2?-3=0 and 3y+2?+2?+7=0 are not parallel, so
they must intersect along a line that is common to both of them.
The vector parametric equation is:

Consider the following planes.
x + y + z = 1, x + 3y + 3z = 1
(a) Find parametric equations for the line of intersection of
the planes. (Use the parameter t.)
(x(t), y(t), z(t)) =
(b) Find the angle between the planes. (Round your answer to one
decimal place.)
°

a= 1, b =6 , c= 4
T(x,y,z) =(x^2 )y +(y^2 )z
Find the directional derivative of T at the point (a − 1, a, a +
1) along
the direction of the vector < a, b, c >.

the region enclosed by the cylinder x^2+y^2=64 and the planes
z=0 and x+y+z=16

3. Consider the following two lines:
x = c + t, y = 1 + t, z = 5 + t and x = t, y = 1 - t, z = 3 +
t.
Is there a value c that makes the two lines intersect? If so,
find it. Otherwise, give a reason.
4. A particle starts at the origin and moves along the shortest
path to the line determined by the two points P =(1,2,3) and Q
=(3,-2,-1)....

f(x, y, z) =
xe4yz, P(1, 0, 3),
u = <2/3, -1/3, 2/3>
(a) Find the gradient of f.
∇f(x, y, z) =
< , , >
(b) Evaluate the gradient at the point P.
∇f(1, 0, 3) = < , ,
>
(c) Find the rate of change of f at P in the
direction of the vector u.
Duf(1, 0, 3) =

. Find the intersection of the planes x − y + 5z = 9 and
x = 1 + s − t
y = 1 +2 s − t
z = 2 − s + t .
(a) Find the line ℓ1 perpendicular to the first of these planes
and passing across the point (1, 2, 2).
(b) Find a line ℓ2 perpendicular to the second of these planes
and passing across the point (1, 2, 2).
(c) Find the...

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