prove that normal to plane containing 3 points whose position vectors are a vector,b vector,c vectorlies in direction addition of cross product of vectors b and c and cross product of vectors c and a and cross product of vectors a and b
To prove that the normal to a plane containing three points with position vectors a, b, and c lies in the direction of the addition of the cross products of vectors b and c, c and a, and a and b, we need to follow these steps:
Step 1: Calculate the vectors between the given points. Let's denote the vector between points a and b as AB, between points b and c as BC, and between points c and a as CA.
AB = b - a
BC = c - b
CA = a - c
Step 2: Calculate the cross products of vectors AB, BC, and CA.
AB × BC = (b - a) × (c - b)
BC × CA = (c - b) × (a - c)
CA × AB = (a - c) × (b - a)
Step 3: Calculate the sum vector of the cross products.
Sum = (AB × BC) + (BC × CA) + (CA × AB)
Step 4: Determine if the normal vector to the plane containing points a, b, and c is parallel to the sum vector.
To prove this, we need to show that the dot product of the normal vector and the sum vector is equal to zero.
Normal ⋅ Sum = 0
If the dot product is zero, it means that the vectors are perpendicular, thus proving that the normal to the plane containing points a, b, and c lies in the direction of the sum vector.