# Determine a direction vector when a line perpendicular and a point is given

• euro94
In summary, the line perpendicular to the line r=[-1,0,1] + t [2,3,-2] and passing through the point (-7,-9,7) has a direction vector of [-7,-9,7]. The final equation for the line is r=[-1,0,1] + 3 [2,3,-2]. The question of finding "the line perpendicular" is invalid since there are infinitely many lines that can be perpendicular to the given line in three dimensions.
euro94
Find the direction vector of the line perpendicular to the line r=[-1,0,1] + t [2,3,-2] and passing through the point (-7,-9,7)

welcome to pf!

hi euro94! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

I tried finding the dot product of the line's direction vector with a vector from (-7,9,7). I tried (-1+2t, 3t, 1-2t) - (-7,-9,7) and I dotted it with (2,-3,2) and made it equal to zero, and then i got stuck :(

euro94 said:
I tried finding the dot product of the line's direction vector with a vector from (-7,9,7). I tried (-1+2t, 3t, 1-2t) - (-7,-9,7) and I dotted it with (2,-3,2) and made it equal to zero, and then i got stuck :(

(you mean (2,3,-2)?)

that should work …

what did you get?​

I got (6+2t,3t+9,6-2t) and i dotted it with (2,3,-2) and then i made it equal to zero and i got 12+4t+9t+27-12+4t and i solved for t and i got -27/17, but I'm not sure what to do next..

i mean i got a value of -27/14 for t, i had a miscalculation, but I am not sure what step to take next

hi euro94!

(just got up :zzz:)
euro94 said:
i mean i got a value of -27/14 for t, i had a miscalculation, but I am not sure what step to take next

that value of the parameter (t) gives you the foot of the perpendicular …

so put it into [-1,0,1] + t [2,3,-2] to give you the actual coordinates,

and then join that to [-7,-9,7]

oppsss, i got t=-3 and i plugged it into the equation and i got that my direction vector is [-7,-9,7] and so my overall equation is r=[-7,-9,7] + t[-7,-9,7]

is that normal?

you mean it goes through the origin?

can you please write it all out in one go, if you want it checked?

okay :)
(-1+2t, 3t, 1-2t) - (-7,-9,7) = (6+2t, 3t+9, -6-2t)
dot product:
(6+2t, 3t+9, -6-2t) dot (2,3,-2) =0
12+4t+9t+27+12+4t =0
51+17t=0
-51/17=t
-3=t
r=[-1,0,1] + 3 [2,3,-2]
= [-7,-9,7]

is that right? :)

euro94 said:
okay :)
(-1+2t, 3t, 1-2t) - (-7,-9,7) = (6+2t, 3t+9, -6-2t)
dot product:
(6+2t, 3t+9, -6-2t) dot (2,3,-2) =0
12+4t+9t+27+12+4t =0
51+17t=0
-51/17=t
-3=t
r=[-1,0,1] + 3 [2,3,-2]
= [-7,-9,7]

hmm … that's weird!

looking back at the original question now, obviously [-7,-9,7] does actually lie on the original line …

so (since we're in three dimensions), asking for "the line perpendicular" makes no sense, since there's an infinite number of them!

thank youu :)

## 1. What is a direction vector?

A direction vector is a mathematical representation of the direction and magnitude of a line or a vector. It is typically expressed as a set of coordinates, such as (x,y,z) or i1j1k1.

## 2. How do you find a direction vector when a line perpendicular and a point is given?

To find a direction vector when a line perpendicular and a point is given, you can use the cross product of the vector representing the perpendicular line and the vector connecting the given point to a point on the line. This will give you a direction vector that is perpendicular to the given line.

## 3. What is the significance of a direction vector?

A direction vector is important in mathematics and physics as it allows us to determine the direction and orientation of a line or a vector. It is also used in various applications such as computer graphics, navigation, and optimization problems.

## 4. Can a direction vector be negative?

Yes, a direction vector can have negative components. This simply indicates the direction and magnitude of the vector in the opposite direction. For example, if a direction vector is (-2,3), it means that the vector is pointing in the negative x-direction with a magnitude of 2 and in the positive y-direction with a magnitude of 3.

## 5. Is a direction vector unique?

Yes, a direction vector is unique for a given line or vector. This is because it represents the direction and orientation of the line or vector and is not affected by its magnitude. However, there may be multiple direction vectors that are perpendicular to a given line.

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