Question

Verify that the line with equation
r=2i +4j +k +t(-4i+4j-5k) lies wholly in the plane with equation 3x-2y+4z=2

Answer 1

wholly?

Answer 2

Yeah, Wholly or Fully

Answer 3

is there some theorem we can use?

Answer 4

can you give me a point on vector r

Answer 5

That's all it gives.

Answer 6

the vector normal to plane is <3,-2,4>
if you can show that this is perpendicular to line vector then they are going same direction

finally you have to show they then go through the same point
look at t=0
r = 2i+4j+k, which is point (2,4,1)
the plane also goes through this point --> 3*2 -2*4+4*1 = 2

Answer 7

Why does t=0?

Answer 8

isn't r = 2i+4j+k supposed to be position vector passing through 0,0,0 ?

Answer 9

to show 2 vectors are perpendicular, the dot product will be 0
<-4,4,-5> * <3,-2,4> = -40 ??
hmm is that maybe a positive 5?

Answer 10

No. It's negative... I thought it would be too... so that it could be 2=2?

Answer 11

if the line is wholly in the plane, then wouldn't all the points for all t be on the plane?

Answer 12

yes

Answer 13

@dumbcow isn't r supposed to be position vector passing through the origin??

Answer 14

yeah i guess..oh i am interpreting it like its a line equation

Answer 15

well would you agree, you could say

x = 2-4t
y=4+4t
z=1-5t

Answer 16

if it is then ... the given plane never passes through the origin and position vector passes through the origin

moreover, when i studied 3d geometry, i studied line as

(x-x1)/l = (y-y1)/n = ... k