Write down an equation for the plane passing through the points [2,0,0][2,0,0], [−3,3,−2][−3,3,−2], and [2,−1,1][2,−1,1]. For example: x+y+z=1.
Question
Write down an equation for the plane passing through the points [2,0,0][2,0,0], [−3,3,−2][−3,3,−2], and [2,−1,1][2,−1,1]. For example: x+y+z=1.
Solution
The equation of a plane in 3D space is given by the formula:
A(x - x1) + B(y - y1) + C(z - z1) = 0
where (x1, y1, z1) is a point on the plane and (A, B, C) is the normal vector to the plane.
The normal vector can be found by taking the cross product of two vectors that lie in the plane. We can get these two vectors by subtracting the coordinates of the given points:
Vector1 = [2,0,0] - [-3,3,-2] = [5,-3,2] Vector2 = [2,0,0] - [2,-1,1] = [0,1,-1]
Now, we calculate the cross product of these two vectors to get the normal vector:
Normal vector = Vector1 x Vector2 = [(-3*-1) - (21), (20) - (5*-1), (51) - (-30)] = [3, 5, 5]
So, the equation of the plane is:
3(x - 2) + 5(y - 0) + 5(z - 0) = 0
Simplifying this gives:
3x + 5y + 5z = 6
So, the equation of the plane passing through the points [2,0,0], [-3,3,-2], and [2,-1,1] is 3x + 5y + 5z = 6.
Similar Questions
Which of the following is a plane passing through the points [1,−1,3][1,−1,3], [2,−1,1][2,−1,1], and [0,3,−14][0,3,−14]?Please select all correct answers.[1,−1,3]+s[2,−1,1]+t[−2,4,−15][1,−1,3]+𝑠[2,−1,1]+𝑡[−2,4,−15][1,−1,3]+s[1,0,−2]+t[−1,4,−17][1,−1,3]+𝑠[1,0,−2]+𝑡[−1,4,−17][1,−1,3]+s[1,0,−2]+t[−2,4,−15][1,−1,3]+𝑠[1,0,−2]+𝑡[−2,4,−15][1,−1,3]+s[2,−1,1]+t[0,3,−14]
Let P𝑃 be the plane passing through the points [1,−2,−1][1,−2,−1], [−2,3,−3][−2,3,−3], and [2,0,3][2,0,3].Where does the line [1,3,1]+s[2,−2,0][1,3,1]+𝑠[2,−2,0] intersect this plane?
Find the equation of the plane through the point p=(3,2,3),and paralled to the plane y-x+2z = -1.
Find an equation of the plane. the plane that passes through the point (3, 6, −1) and contains the line x = 4 − t, y = 2t − 1, z = −3t
The plane passing through the origin and containing the lines whose direction cosines are proportional to 1,-2,2 and 2,3,-1 passes through the point
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.