Soru
d. Find a vector equation and the cartesian equation of the perpendicular bisectors of PQ, where P and Q are the points with position vectors; i) -3i-j and 7i+j (4marks) ii) ai+bj and 2ai+3bj (3marks)
Çözüm
4.5
(256 Oylar)
Behçet
Usta · 5 yıl öğretmeni
Uzman doğrulaması
Cevap
To find the vector equation and the Cartesian equation of the perpendicular bisector of PQ, we need to follow these steps:1. Find the midpoint of PQ.2. Find the direction vector of the perpendicular bisector.3. Use the midpoint and direction vector to find the vector equation.4. Convert the vector equation to the Cartesian equation.Let's solve each part of the problem:i) For points P(-3i-j) and Q(7i+j):1. Midpoint of PQ:Midpoint = ((-3i-j) + (7i+j))/2 = (4i, 0)2. Direction vector of the perpendicular bisector:The direction vector is perpendicular to PQ, so it is given by the cross product of the position vectors of P and Q:Direction vector = (-3i-j) x (7i+j) = (-3, -1) x (7, 1) = (1, 3)3. Vector equation of the perpendicular bisector:The vector equation of a line passing through the midpoint (4i, 0) and having the direction vector (1, 3) is given by:r = (4i, 0) + t(1, 3) = (4+t)i + 3tj4. Cartesian equation of the perpendicular bisector:To convert the vector equation to the Cartesian equation, we can eliminate the parameter t:(4+t)i + 3tj = (4i, 0)4+t = 4t = 00 = 3tTherefore, the Cartesian equation of the perpendicular bisector is x = 4.ii) For points P(ai+bj) and Q(2ai+3bj):1. Midpoint of PQ:Midpoint = ((ai+bj) + (2ai+3bj))/2 = (3ai+2bj)2. Direction vector of the perpendicular bisector:The direction vector is perpendicular to PQ, so it is given by the cross product of the position vectors of P and Q:Direction vector = (ai+bj) x (2ai+3bj) = (a, b) x (2a, 3b) = (3b-a, -2a+3b)3. Vector equation of the perpendicular bisector:The vector equation of a line passing through the midpoint (3ai+2bj) and having the direction vector (3b-a, -2a+3b) is given by:r = (3ai+2bj) + t(3b-a, -2a+3b) = (3ai+2bj+3bt-ataj, -2at+3bt)4. Cartesian equation of the perpendicular bisector:To convert the vector equation to the Cartesian equation, we can eliminate the parameter t:(3ai+2bj+3bt-ataj, -2at+3bt) = (3ai+2bj, 0)3ai+2bj+3bt-ataj = 3ai+2bj3bt-ataj = 0t(3b-a) = 0Therefore, the Cartesian equation of the perpendicular bisector is 3b-a = 0 or a = 3b.