1.1.58 Vector Algebra

condition: Given two geometric vectors \( \bar{P} \) and \( \bar{Q} \). Represent the vector \( \bar{P} \) as the sum of two vectors \( \overline{P_{1}} \) and \( \overline{P_{2}} \) such that the vector \( \overline{P_{1}} \) is perpendicular to the vector \( \bar{Q} \), and the vector \( \overline{P_{2}} \) is collinear to the vector \( \bar{Q} \). \[ \bar{P}(2,-4,12), \quad \bar{Q}(1,-1,-5) \]