On a vector, the maximum value of the product of two vectors. (related to angle)

It is known in Rt△ABC that ∠ A = 90 and BC = A. If the line segment PQ with a length of 2a takes point A as the midpoint, what is the included angle between vector PQ and vector BC, and the value between vector BP and vector CQ is the largest? Find this maximum value. The description vector AB is labeled as "AB"

With A as the origin and the rays of AB and AC as the positive directions of X axis and Y axis, a (0,0), b (c,0), c (0,b), P(p, q), Q(-p, -q), obviously, b? +c？ =a？ ① p？ +q？ =a？ ②. "PQ" = (-2p, -2q) and "BC" = (-c, b). If the included angle between "PQ" and "BC" is θ, then COSθ = "PQ" BC/[| PQ ||| BC |] = (2pc-2bc |). ) +(pc-bq), from ② ③: "BP" CQ "=-a? +a？ cosθ=a？ (cosθ- 1) So when θ = 90, "BP" and "CQ" get the maximum value of 0.