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Find a vector that is perpendicular to the plane passing through the three given points.
Full solution
Q. Find a vector that is perpendicular to the plane passing through the three given points.
- Find Vectors and : To find a vector that is perpendicular to the plane defined by three points, we can use the cross product of two vectors that lie in the plane. We first need to find two vectors that are based on the given points. We can do this by subtracting the coordinates of the points to get the vectors and .
- Calculate Vector PQ: Calculate the vector PQ by subtracting the coordinates of point Q from the coordinates of point P.PQ =
- Calculate Vector PR: Calculate the vector PR by subtracting the coordinates of point R from the coordinates of point P.
- Find Cross Product: Now we will find the cross product of vectors and to get a vector that is perpendicular to the plane.The cross product of two vectors and is given by the determinant of the following matrix:
- Calculate Cross Product PQ x PR: Calculate the cross product using the determinant method:\begin{vmatrix} i & j & k \ 1 & 1 & -1 \ -3 & 0 & 0 \ \end{vmatrix}The cross product is:
