# Parametric Equations: Find Line P Passing Through P

• yazz912
In summary, the problem is to find the equation of a line passing through a given point and parallel to the line of intersection of two planes. The solution involves finding the cross product of the normal vectors of the planes, which gives the direction vector for the line. The cross product is then simplified and used to create the parametric equations for the line.
yazz912
1. The problem statement, all variables and given/known

Find the equation of a line passing through the point P=(-1,2,3) that is parallel to the line of intersection of the planes 3x-2y+z=4 and x+2y+3z=5 . Express your answer in parametric equations .

2. Homework Equations

Cross product of normal vectors
N1 X N2

3. The Attempt at a Solution
Normally all other problems we've had to use cross product of the normals vectors of planes.
n1= <3,-2,1>
n2= <1,2,3>

Cross product I get < -8,-8,8>

Idk what my next step is since I am trying to find a line passing through point P PARALLEL to line of intersection?

The vector you get from the cross product gives you your direction vector for the line -- it points in the direction of the line. Two lines are parallel if they have the same direction vectors (or really if their direction vectors are parallel). To find your line, assign it the vector of <-8, -8, 8> (or <-1, -1, 1> if you want to reduce it)

yazz912 said:
1. The problem statement, all variables and given/known

Find the equation of a line passing through the point P=(-1,2,3) that is parallel to the line of intersection of the planes 3x-2y+z=4 and x+2y+3z=5 . Express your answer in parametric equations .

2. Homework Equations

Cross product of normal vectors
N1 X N2

3. The Attempt at a Solution
Normally all other problems we've had to use cross product of the normals vectors of planes.
n1= <3,-2,1>
n2= <1,2,3>

Cross product I get < -8,-8,8>

Idk what my next step is since I am trying to find a line passing through point P PARALLEL to line of intersection?

Can't you use your cross product vector for a direction vector for your line?

hi yazz912!
yazz912 said:
Cross product I get < -8,-8,8>

looks fine so far

ok, you're only interested in the direction, so you may as well call that < -1,-1,1>, or even slightly better still < 1,1,-1> …

so what is the parametric equation of the line through P parallel to <1,1,-1> ?

So using my point P=(-1,2,3)
And my direction vector of <-1,-1,1>

Would my parametric equation be

x= -1t -1
y= -1t +2
z= 1t +3

yup!

Ok thanks!:)

## 1. What are parametric equations?

Parametric equations are a set of equations that express a quantity in terms of one or more independent variables, known as parameters. These equations are commonly used in mathematics and physics to describe curves and surfaces.

## 2. How do you find a line passing through a point using parametric equations?

To find a line passing through a point using parametric equations, you first need to determine the slope of the line. This can be done by finding the difference in the y-coordinates and the difference in the x-coordinates between the given point and another point on the line. Then, use the slope and the given point to write the parametric equations in terms of a parameter, such as t or s.

## 3. Can parametric equations be used to represent straight lines?

Yes, parametric equations can be used to represent straight lines. This can be done by setting the parametric equations for the x and y coordinates equal to a linear function, such as y = mx + b, where m is the slope and b is the y-intercept.

## 4. What is the advantage of using parametric equations to represent lines?

One advantage of using parametric equations to represent lines is that they can easily be extended to represent curves and surfaces. This allows for a more flexible and comprehensive approach to solving mathematical problems.

## 5. Can parametric equations be graphed?

Yes, parametric equations can be graphed. Each parameter in the equations represents a point on the graph, and by varying the parameter, you can create a series of points that trace out a curve or surface. These points can then be plotted to create a visual representation of the parametric equations.

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