Question

Determine the distance between P(3 , -2 , 5) and the plane 2x + 4y - z = 2.

Please show all work. Thanks (:

Answer 1

2x + 4y - z = 2
this is plane equation in R3
(2,4,-1) it's the perpendicular vector to this plane
the corresponding unit vector is (2,4,-1)/sqrt21
the point P(3 , -2 , 5) is
since 2*3+4*(-2)+5*(-1)=6-8-5=-7 not equals 2
distance is -7-2/sqrt21=-9/sqrt21

Answer 2

So the answer is : -7-2/sqrt21=-9/sqrt21
=-1.96?

Answer 3

would be positive. The - sign means that is on the side of the plane which doen's contain the origin. But i don't think you need that. So just use +

Answer 4

But how did you even solve up to that point? I dont understand what you did

Answer 5

....

Answer 6

i explain you for the case of the line in R2. In R3 is the same. n is the unit normal vector to the line and r is the position vector. So the equation of the line in vector form is: r.n=c, where c is constant. What this equation states, that the projection on the unit normal of the position vector of the points belonging to the line is constant. You can folow the rest from here

Answer 7

got it?

Answer 8

What is R2 and R3?

Answer 9

spaces: 2 dimention, 3 dimention of the real numbers pairs, triplets

Answer 10

I have never dealt with this, I am in 12 Calculus

Answer 11

no idea what is 12 Calculus

Answer 12

Grade 12 Calculus

Answer 13

you mean school?

Answer 14

Yes

Answer 15

highschool

Answer 16

Yes

Answer 17

ok, more easy way:

Answer 18

find any point Q on the plane. It means 3 numbers(x,y,z) matching the 2x + 4y - z = 2
for example (0,0,-2). Take vector PQ and do dot product with perpendicular vector to the plane, (2,4,-1) devided by magnitude of perpendicular vector , sqrt21. This is the distanece

Answer 19

How do you get 21?

Answer 20

if v(x,y,z) is a vector, it's length is:
\[|v|=\sqrt{x ^{2}+y ^{2}+z ^{2}}\]

Answer 21

ok

Answer 22

Can you take me step-by-step on how you get the answer?

Answer 23

Im trying to pretend its like a test question, so my teacher would mark my work

Answer 24

**work shown

Answer 25

PQ=(3,-2,7)
(3,-2,7).(2,4,-1)=6-8-7=-9
deviding by sqrt21 you get -9/sqrt21

Answer 26

So if I show that work, I would get full marks?

Answer 27

i guess, if this is the way you supose to do it, :)

Answer 28

no idea about highschool way...

Answer 29

What methods are you using to solve this question?

Answer 30

length of vector
dot product

Answer 31

These are the chapter titles: Cartesion vectors, dot product, Applications of the dot product, Vectors in three space, the cross product and its properties and applications of the dot prosuct and cross product

Answer 32

so then this is the way

Answer 33

What are the methods you used?

Answer 34

PQ=(3,-2,7) isnt this supposed to be (3, -2, 5)?