The equation of a plane through the origin and perpendicular to the vector (1, -2, 5) is derived using the formula for a plane in three-dimensional space. The relevant equation is given by \( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \), where \( (x_0, y_0, z_0) \) is a point on the plane and \( (a, b, c) \) are the components of the normal vector. Substituting the origin (0, 0, 0) and the normal vector (1, -2, 5) results in the equation \( 1(x - 0) - 2(y - 0) + 5(z - 0) = 0 \), simplifying to \( x - 2y + 5z = 0 \).
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