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]

To find the equation of a plane passing through three points, we first need to find two vectors that are in the plane. We can do this by subtracting the coordinates of the points from each other.

Let's denote the points as follows: P1 = [1, -1, 3] P2 = [2, -1, 1] P3 = [0, 3, -14]

We can find two vectors in the plane by subtracting the coordinates of P1 from P2 and P3:

Vector1 = P2 - P1 = [2-1, -1-(-1), 1-3] = [1, 0, -2] Vector2 = P3 - P1 = [0-1, 3-(-1), -14-3] = [-1, 4, -17]

The equation of a plane can be written in the form P = P1 + sVector1 + tVector2, where s and t are parameters. Substituting the vectors we found into this equation gives:

P = [1, -1, 3] + s*[1, 0, -2] + t*[-1, 4, -17]

So, the correct answer is [1, -1, 3] + s*[1, 0, -2] + t*[-1, 4, -17].

To find the plane passing through the points [1,−1,3], [2,−1,1], and [0,3,−14], we first need to find two vectors that are in the plane. We can do this by subtracting the coordinates of the points from each other.

Let's denote the points as A[1,−1,3], B[2,−1,1], and C[0,3,−14].

Vector AB = B - A = [2-1, -1-(-1), 1-3] = [1, 0, -2] Vector AC = C - A = [0-1, 3-(-1), -14-3] = [-1, 4, -17]

The equation of a plane in 3D space given a point (x0, y0, z0) and two vectors (a, b, c) and (d, e, f) in the plane is:

(x - x0)a + (y - y0)b + (z - z0)c = 0 (x - x0)d + (y - y0)e + (z - z0)f = 0

Substituting the coordinates of point A and the components of vectors AB and AC into the equations, we get:

(x - 1)1 + (y - (-1))0 + (z - 3)(-2) = 0 (x - 1)(-1) + (y - (-1))4 + (z - 3)(-17) = 0

Simplifying these equations, we get:

x - 2z - 5 = 0 -x + 4y - 17z + 6 = 0

So, the plane passing through the points [1,−1,3], [2,−1,1], and [0,3,−14] is represented by the equations x - 2z - 5 = 0 and -x + 4y - 17z + 6 = 0.

Looking at the options given, the correct answer is [1,−1,3]+s[1,0,−2]+t[−1,4,−17].