Finding Vector Equation

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In summary, the problem is asking for a vector equation with parameter T that passes through the point (6,0,1) and is perpendicular to the plane x+4y+6z = 5. One way to approach this is by finding the direction numbers n(a,b,c) for a direction that is perpendicular to the given plane. This can be done by taking the dot product of the direction vector <1,4,6> and the normal vector of the plane <1,4,6>. Once the direction and a point on the line are known, it is possible to find the equation of the line.
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Homework Statement


Find the vector equation with parameter T that passes through the point (6,0,1) and is perpendicular to the plane x+4y+6z = 5






The Attempt at a Solution


So I'm looking for the direction numbers n (a,b,c) for whatever direction is perpendicular to the plane given x+4y+6z = 5

What if I try to take the dot product of i+4j+6k with xi+yj+zk and solve for when it equals zero. The problem with doing that is there are more than one possible answers that solve that. Also since x+4y+6z = 5 is the linear equation of a plane I'm not even sure I'm thinking about this right at all X_x
 
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PsychonautQQ said:

Homework Statement


Find the vector equation with parameter T that passes through the point (6,0,1) and is perpendicular to the plane x+4y+6z = 5






The Attempt at a Solution


So I'm looking for the direction numbers n (a,b,c) for whatever direction is perpendicular to the plane given x+4y+6z = 5

What if I try to take the dot product of i+4j+6k with xi+yj+zk and solve for when it equals zero. The problem with doing that is there are more than one possible answers that solve that. Also since x+4y+6z = 5 is the linear equation of a plane I'm not even sure I'm thinking about this right at all X_x

One direction vector for the line is <1, 4, 6>, which I got by inspection. A normal to the plane Ax + By + Cz = D is the vector <A, B, C>. Once you know the direction of a line and a point on it, it's straightforward to get the equation of the line.
 

1. What is a vector equation?

A vector equation is a mathematical representation of a line or plane in three-dimensional space using vectors. It is often written in the form of r = r0 + tv, where r0 is a fixed vector and v is a direction vector.

2. How is a vector equation different from a parametric equation?

A parametric equation represents a curve or surface by using one or more parameters, while a vector equation represents a line or plane using vectors. In other words, a parametric equation describes the coordinates of points on a curve or surface, while a vector equation describes the position of points on a line or plane.

3. How do you find the vector equation of a line given two points?

To find the vector equation of a line given two points, first find the direction vector by subtracting the coordinates of one point from the coordinates of the other point. Then, choose one point and use it as the r0 in the equation r = r0 + tv, where t is a parameter and v is the direction vector.

4. Can a vector equation represent a line or plane in two-dimensional space?

No, a vector equation can only represent a line or plane in three-dimensional space. In two-dimensional space, a line can be represented by a single parametric equation while a plane can be represented by two parametric equations.

5. How can a vector equation be used to find the distance between a point and a line?

To find the distance between a point and a line using a vector equation, first find a vector connecting the given point to any point on the line. Then, find the dot product of this vector and the direction vector of the line. Finally, divide the absolute value of the dot product by the magnitude of the direction vector to find the distance.

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