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Answer: a
Explanation: The cross product of p1, p2, p3 is a X b = -j – k and its magnitude is 1.414.
The unit normal vector is given by, (-j –
k)/1.414.
To find the unit vector to the points p1(0,1,0), p2(1,0,1), and p3(0,0,1), we first need to determine a direction vector by subtracting these points. A common approach is to find the centroid of the three points.
1. Find the centroid (C):
[
C = left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3}, frac{z_1 + z_2 + z_3}{3} right) = left( frac{0 + 1 + 0}{3}, frac{1 + 0 + 0}{3}, frac{0 + 1 + 1}{3} right) = left( frac{1}{3}, frac{1}{3}, frac{2}{3} right)
]
2. Find the direction vectors from the centroid to each point:
– From C to p1:
[
p1 – C = left( 0 – frac{1}{3}, 1 – frac{1}{3}, 0 – frac{2}{3} right) = left( -frac{1}{3}, frac{2}{3}, -frac{