Mathematical problems about vectors In △ ABC, satisfy: ab ⊥ AC, M is the midpoint of BC (1) If | ab | = | AC |, find the cosine of the angle between vector AB + 2Ac and vector 2Ab + AC; (2) If O is any point on the line am, and | ab | = | AC |, = root 2, find the minimum value of OA * ob + OC * OA (3) If P is a point on BC, and AP = 2, AP * AC = 2AP * AB = 2, find the minimum value of | AB + AC + AP |

Mathematical problems about vectors In △ ABC, satisfy: ab ⊥ AC, M is the midpoint of BC (1) If | ab | = | AC |, find the cosine of the angle between vector AB + 2Ac and vector 2Ab + AC; (2) If O is any point on the line am, and | ab | = | AC |, = root 2, find the minimum value of OA * ob + OC * OA (3) If P is a point on BC, and AP = 2, AP * AC = 2AP * AB = 2, find the minimum value of | AB + AC + AP |

Where is the problem (I) (vector AB + 2Ac) * (vector 2Ab + AC) = 4 | ab | square | vector AB + 2Ac | = root 5 | ab | = | vector 2Ab + AC | cos angle = 0.8
(2) OA * ob + OC * OA = vector 2oa * om = - 2oa * ob | let | OA | = x get BC = 2 = 2om from ab = | AC |, = root 2, so when om = 1-x OA * ob + OC * OA = - 2x (1-x) x = 1 / 2, the minimum OA * ob + OC * OA = - 1 / 2
(3) Let AB = x AC = y | AB + AC + AP | = the square of the root sign vector AB + AC + AP = the root sign (x * 2 + y * 2 + 10) = the root sign (45 / 4 + Tana square + 1 / 4 Tana Square) > = the root sign 49 / 4
The minimum value of | AB + AC + AP | is 7 / 2