# Equation of a line perpendicular to two vectors

• morsel
The symmetric equation should be x = y+1 = z-1. In summary, the parametric and symmetric equations for the line through the point (2,1,0) and perpendicular to both <1,1,0> and <0,1,1> are x = 2 + t, y = 1 - t, z = t and x = y+1 = z-1, respectively.
morsel

## Homework Statement

Find the parametric and symmetric equations for the line through the point (2,1,0) and perpendicular to both <1,1,0> and <0,1,1,>

Parametric equ:
x = x_0 + at
y = y_0 + bt
z = z_0 + ct

## The Attempt at a Solution

Cross product of <1,1,0> and <0,1,1> = <1,-1,1> = <a,b,c>

Parametric equation:
x = 2 + t
y = 1 - t
z = t

Symmetric equation:
x-2 = y-1 = z

I'm not sure if I did this correctly. Is the cross product of the two vectors <a,b,c>?

Looks pretty much OK. Your cross product and parametric equations are correct. Check your signs on the symmetric version.

Thanks for catching that.

## What is the equation of a line perpendicular to two vectors?

The equation of a line perpendicular to two vectors is given by the cross product of the two vectors. This means that the dot product of the resulting line with either of the two vectors will be equal to zero.

## How do you find the equation of a line perpendicular to two vectors?

To find the equation of a line perpendicular to two vectors, you first need to find the cross product of the two vectors. This will give you a vector that is perpendicular to both of the original vectors. Then, you can use this vector and any point on the line to create the equation of the line in the form of ax + by + c = 0.

## Can a line be perpendicular to more than two vectors?

No, a line can only be perpendicular to two vectors at a time. If you have more than two vectors, you can find the equation of a line perpendicular to any two of them by using the cross product method.

## What are the properties of a line perpendicular to two vectors?

A line perpendicular to two vectors will have the following properties:

• It will be perpendicular to both of the original vectors.
• The dot product of the line with either of the two vectors will be equal to zero.
• The slope of the line will be the negative reciprocal of the slope of the original vectors.

## Can the equation of a line perpendicular to two vectors be written in different forms?

Yes, the equation of a line perpendicular to two vectors can be written in different forms depending on the given information. It can be written in the standard form ax + by + c = 0, the slope-intercept form y = mx + b, or the point-slope form y - y1 = m(x - x1). The form of the equation will depend on the given information and what is most convenient for the problem at hand.

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