Determine an equation for the line parallel to 2x + 6y + 4z = 1 and contains the point P(3, 2, 1).
I think you posted this before, and probably got no response.
You asked for "the" line parallel to the plane
2x+6y+4z = 1.
There is no unique line.
Think of the floor of your kitchen as 2x+6y+4z = 1 and you have a table whose top is parallel to the floor and contains the point (3,2,1).
Laying down a ruler would be a line parallel to the plane and passing through the point.
How many such positions could you choose ?
a normal to your given plane is (2,6,4) or (1,3,2)
so any vector whose dot product with the above normal would be parallel to the plane
e.g. (-5,1,1)∙(1,3,2) = -5+2+3 = 0
and (1,-1,1)∙(1,2,3) = 1-3+2 = 0 are just two such vectors
let's take (-5,1,1)
so the equation of one such line is
x = 3 -5t
y = 2 + t
z = 1 + t
thank you :)
To find an equation for the line parallel to a given plane and passing through a given point, we need to find the normal vector of the plane and then use it to construct the equation.
First, let's write the given equation of the plane in the standard form Ax + By + Cz + D = 0, where A, B, C are the coefficients of the variables x, y, z respectively, and D is a constant.
In this case, the given equation is 2x + 6y + 4z = 1.
To determine the normal vector of the plane, we'll identify the coefficients A, B, and C.
From the equation 2x + 6y + 4z = 1, we can see that A = 2, B = 6, and C = 4.
Now, since we want a line parallel to the plane, the direction vector of the line will be the same as the normal vector of the plane.
Therefore, the direction vector of the line is <2, 6, 4>.
Using the point-slope form of the equation of a line, we have:
(x - x₁) / a = (y - y₁) / b = (z - z₁) / c,
where (x₁, y₁, z₁) represent the coordinates of the given point P(3, 2, 1), and (a, b, c) form the direction vector for the line.
Substituting the values, the equation becomes:
(x - 3) / 2 = (y - 2) / 6 = (z - 1) / 4.
Alternatively, if you prefer the slope-intercept form of the equation of a line, you can rearrange the equation as follows:
(x - 3) / 2 = (y - 2) / 6 = (z - 1) / 4 = t,
where t is a variable representing a parameter.
This form allows you to easily calculate points on the line by assigning specific values to t.