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# Calculus- distance from a point watch

1. Q: calculate the distance between point P and the given line:

P(1,2,-1) and line r=[1,0,0] + s [2,-1,2] ,sER

A: So, first I found AP (OP-OA) and then

I used the formula

d= | AP X m |
------------
|m|

and got
d= |[3,-2,-4] |
----------
sqrt. 10
But the answer in the back of the book is 1.80 and I'm confused as to how they got that because I have tried dividing the vector by the square root of 10 and I don't think that is allowed. How do they get 1.80 and is the answer I got in an incorrect notation?
2. (Original post by calu)
Q: calculate the distance between point P and the given line:

P(1,2,-1) and line r=[1,0,0] + s [2,-1,2] ,sER

A: So, first I found AP (OP-OA) and then

I used the formula

d= | AP X m |
------------
|m|

and got
d= |[3,-2,-4] |
----------
sqrt. 10
But the answer in the back of the book is 1.80 and I'm confused as to how they got that because I have tried dividing the vector by the square root of 10 and I don't think that is allowed. How do they get 1.80 and is the answer I got in an incorrect notation?
Are you trying to find the shortest distance from the point to the line or are you given a value of 's'?
3. Remember that the shortest distance is the perpendicular distance. Consider the point of intersection of the line and the perpendicular through P. Remember that this point of intersection lies on the given line. What can you say about the direction vectors of the given line and the perpendicular through P?
4. (Original post by Mr M)
As it happens neither you nor the book are right.

The correct distance is
How is that obtained?
5. (Original post by Ateo)
How is that obtained?
Assuming this is the shortest distance => a.b = 0
Work out a.b CORRECTLY
Work out the magnitude of , where Q is the new point found.
6. Edit: I have deleted my previous post as 1.8 IS the correct answer. I missed a sign when copying down the question.
7. (Original post by Mr M)
...
I don't see the point of introducing all those variables. Whats the need for making a vector equation of the line through P?
8. (Original post by Ateo)
I don't see the point of introducing all those variables. Whats the need for making a vector equation of the line through P?
I was just being long-winded. Got there now.
9. (Original post by joostan)
Assuming this is the shortest distance => a.b = 0
Work out a.b in terms of s, solve for s this will give you a new point.
Work out the magnitude of , where Q is the new point found.
Spoiler:
Show
Coincidentally Q = (1,0,0)
You made exactly the same sign error as me. Q = (1/9, 4/9, -8/9).
10. (Original post by Mr M)
You made exactly the same sign error as me. Q = (1/9, 4/9, -8/9).
oops

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