Find a vector normal to a plane consisting of points p1(0,1,0), p2(1,0,1) and p3(0,0,1)

To find a vector normal to a plane defined by three points ( p_1(0,1,0) ), ( p_2(1,0,1) ), and ( p_3(0,0,1) ), we can use the following steps:

1. First, find two vectors that lie in the plane by subtracting the coordinates of the points:

[

vec{v_1} = p_2 – p_1 = (1,0,1) – (0,1,0) = (1, -1, 1)

]

[

vec{v_2} = p_3 – p_1 = (0,0,1) – (0,1,0) = (0, -1, 1)

]

2. Next, we need to find the cross product of these two vectors to determine a normal vector to the plane:

[

vec{n} = vec{v_1} times vec{v_2}

]

The cross product can be calculated using the determinant of the matrix:

[

vec{n} = begin{vmatrix}

hat{i} & hat{j} & hat{k} \

1 & -1 & 1 \

0 & -1 & 1