# Finding line perpendicular to 2 other lines in 3D space

• jaejoon89
In summary, to find the equation of a line perpendicular to two given lines, you can use the cross product to determine the direction vector and then find the equation in the general form using a point on the line. In the case of skew lines, the point must be chosen carefully to ensure a unique perpendicular line.
jaejoon89
Find the equations for a line that is perpendicular to both of the following lines

x/3 = y/2 = z/2
x/5 = y/3 = (z-4)/2

---

cross product:
<3,2,2> x <5,3,2> = <-2,4,-1>

then the line is
(x-a)/2 = (b-y)/4 = (z-c)/1

but the answer is supposed to be
(1/2)x - (52/7) = (-1/4)y + (52/21) = z - (208/21)

How on Earth do I get that?

You answer is incorrect because it is not the equation of a single line but rather the general form for a line perpendicular to the two given lines through the point (a, b, c). If the two given lines were either parallel or intersecting, there would exist an infinite number of lines perpendicular to the two given lines and you could pick whatever point (a, b, c) was convenient. However, the two given lines are skew, neither parallel (because the direction vectors <3, 2, 2> and <5, 3, 2> are not parallel) nor intersecting (because the equations x/3= y/2 and x/5= y/3 have x=y= 0 as solution and there is no z satisfying both z/2= 0 and (z-4)/2= 0) so they have a unique perpendicular. You must also determine where that perpendicular crosses one of the lines.

The given answer is the line perpendicular to the two given lines through the point (104/7, 208/21, 208/21)

## 1. How do you determine if two lines in 3D space are perpendicular?

To determine if two lines are perpendicular, you can use the dot product between the direction vectors of the two lines. If the dot product is equal to 0, then the lines are perpendicular. Another way is to calculate the slope of each line and check if they are negative reciprocals of each other.

## 2. What is the formula for finding the line that is perpendicular to two other lines in 3D space?

The formula for finding the line perpendicular to two other lines in 3D space is to take the cross product between the direction vectors of the two lines. This will give you a vector that is perpendicular to both lines, and you can use that to find the equation of the new line.

## 3. Can you find a perpendicular line to two parallel lines?

No, it is not possible to find a line that is perpendicular to two parallel lines in 3D space. This is because parallel lines have the same direction vectors, and the cross product of two identical vectors is always 0, meaning there is no perpendicular vector.

## 4. Is it possible for three lines to be mutually perpendicular in 3D space?

Yes, it is possible for three lines to be mutually perpendicular in 3D space. This means that each line is perpendicular to the other two lines. To determine if three lines are mutually perpendicular, you can use the dot product between each pair of lines. If all three dot products are equal to 0, then the lines are mutually perpendicular.

## 5. What is the significance of finding a line that is perpendicular to two other lines in 3D space?

Finding a line that is perpendicular to two other lines in 3D space is important in many applications, such as engineering and physics. It allows us to determine the shortest distance between two skew lines, find the normal vector of a plane, and solve for the point of intersection between two lines.

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