Finding the Normal Vector and Equation of a Line Perpendicular to 2x + y = 4

In summary: You are looking for a vector that has a dot product of 0 with the normal vector (see also the explanation from the thread "Understanding dot product").In summary, to find the vector equation of a unit normal from the origin to the line 2x + y = 4, you would need to find the normal vector with components 2 and 1, and then use a multiple of this vector to create a dot product of 0 with the normal vector. For part (b), you would need to find another line passing through the point P(0,2) and having a normal vector of <2,1>. This can be done by using the general equation of a line and setting the dot product with the normal
  • #1
imb3cile
1
0
the equation of a straight line is given by 2x + y = 4

(a) find the vector equation of a unit normal from the origin to the line and (b) the equation of a line passing through P(0,2) and normal to 2x + y = 4.


i know i need to use the dot product some how but i am utterly confused as to where to begin! please help.

i think first the vector equation for the given line is r = < 1 , -2 > then i need to dot r with rperp and set equal to 0? i am very confused. thanks!
 
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  • #2
Welcome to PF, imb3cile! :smile:

The general equation of a line is [itex](\vec n, \vec x) = d[/itex]
In this equation [itex]\vec n[/itex] is a normal vector to the line, [itex]\vec x[/itex] is the vector with your coordinates x and y, and d is an arbitrary constant, that creates an offset to the origin.
The parenthesis around the 2 vectors indicate that the dot product is taken.

If the vector n would have components a and b, this would turn out as: ax + by = d

Can you see from your own equation what the normal vector n would be?

For part (a) you would need a multiple of this vector, such that if you fill it in for x and y, it matches the equation.
 

What is a vector equation of a line?

A vector equation of a line is a mathematical representation of a line using vectors, or quantities that have both magnitude and direction. It is typically written as r = a + tb, where r is a position vector, a is the initial point of the line, b is the direction vector, and t is a parameter that represents the distance traveled along the line.

How is a vector equation of a line different from a parametric equation of a line?

A vector equation of a line is similar to a parametric equation of a line, but it uses vectors instead of specific coordinates. This means that a vector equation can represent a line in any direction, while a parametric equation is limited to a specific direction. Additionally, a vector equation is more compact and can easily be converted to other forms, such as a Cartesian equation.

What information do I need to write a vector equation of a line?

To write a vector equation of a line, you need at least two points on the line or the direction vector of the line. If you have two points, you can find the direction vector by subtracting the coordinates of the points. If you have the direction vector, you also need to know a point on the line to determine the initial point, which can be any point on the line.

Can a vector equation of a line represent a line in three dimensions?

Yes, a vector equation of a line can represent a line in three dimensions. In three-dimensional space, the position vector r has three components, and the direction vector b has three components. The equation r = a + tb can represent a line in three dimensions, where a and b are three-dimensional vectors.

How do I convert a vector equation of a line to a Cartesian equation?

To convert a vector equation of a line to a Cartesian equation, you can set the components of the position vector r equal to the corresponding components of a point on the line. Then, you can solve for the parameter t and substitute it into the equation. This will result in a Cartesian equation in terms of x, y, and z.

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