If the distance of point B(2,1,-3) from the line passing through the point A(4,-2,2) , and parallel to the vector c = -4 i - 6 j - 2 k is x , then x^4 + x^2 + 541 =

Step-by-step solution

The position vectors of points A and B are a = 4 i - 2 j + 2 k and b = 2 i + j - 3 k . The vector joining A to B is: AB = b - a = (2-4) i + (1 - (-2)) j + (-3-2) k = -2 i + 3 j - 5 k The line passes through A and is parallel to the vector c = -4 i - 6 j - 2 k . We can take the direction vector of the line as d = 2 i + 3 j + k . Taking the dot product of AB and d : AB d = (-2)(2) + (3)(3) + (-5)(1) = -4 + 9 - 5 = 0 Since AB is perpendicular to the direction of the line, the perpendicular distance x of point B from t