Which of the following is a plane passing through the points [1,1,−2][1,1,−2], [3,4,−3][3,4,−3], and [−2,−1,8][−2,−1,8]?Please select all correct answers.[1,1,−2]+s[2,3,−1]+t[−3,−2,10][1,1,−2]+𝑠[2,3,−1]+𝑡[−3,−2,10][1,1,−2]+s[2,3,−1]+t[−5,−5,11][1,1,−2]+𝑠[2,3,−1]+𝑡[−5,−5,11][1,1,−2]+s[3,4,−3]+t[−5,−5,11][1,1,−2]+𝑠[3,4,−3]+𝑡[−5,−5,11][1,1,−2]+s[3,4,−3]+t[−2,−1,8]
Question
Which of the following is a plane passing through the points [1,1,−2][1,1,−2], [3,4,−3][3,4,−3], and [−2,−1,8][−2,−1,8]?Please select all correct answers.[1,1,−2]+s[2,3,−1]+t[−3,−2,10][1,1,−2]+𝑠[2,3,−1]+𝑡[−3,−2,10][1,1,−2]+s[2,3,−1]+t[−5,−5,11][1,1,−2]+𝑠[2,3,−1]+𝑡[−5,−5,11][1,1,−2]+s[3,4,−3]+t[−5,−5,11][1,1,−2]+𝑠[3,4,−3]+𝑡[−5,−5,11][1,1,−2]+s[3,4,−3]+t[−2,−1,8]
Solution
The equation of a plane passing through three points P1(x1, y1, z1), P2(x2, y2, z2), and P3(x3, y3, z3) can be written in parametric form as:
P = P1 + s(P2 - P1) + t(P3 - P1)
where P is any point on the plane, and s and t are parameters.
Given the points [1,1,-2], [3,4,-3], and [-2,-1,8], we can write the equation of the plane as:
P = [1,1,-2] + s([3,4,-3] - [1,1,-2]) + t([-2,-1,8] - [1,1,-2])
Simplifying, we get:
P = [1,1,-2] + s[2,3,-1] + t[-3,-2,10]
So, the correct answer is [1,1,-2]+s[2,3,-1]+t[-3,-2,10].
Similar Questions
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