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)

To find the vector equation and the Cartesian equation of the perpendicular bisector of PQ, we need to follow these steps:<br /><br />1. Find the midpoint of PQ.<br />2. Find the direction vector of the perpendicular bisector.<br />3. Use the midpoint and direction vector to find the vector equation.<br />4. Convert the vector equation to the Cartesian equation.<br /><br />Let's solve each part of the problem:<br /><br />i) For points P(-3i-j) and Q(7i+j):<br /><br />1. Midpoint of PQ:<br />Midpoint = ((-3i-j) + (7i+j))/2 = (4i, 0)<br /><br />2. Direction vector of the perpendicular bisector:<br />The direction vector is perpendicular to PQ, so it is given by the cross product of the position vectors of P and Q:<br />Direction vector = (-3i-j) x (7i+j) = (-3, -1) x (7, 1) = (1, 3)<br /><br />3. Vector equation of the perpendicular bisector:<br />The vector equation of a line passing through the midpoint (4i, 0) and having the direction vector (1, 3) is given by:<br />r = (4i, 0) + t(1, 3) = (4+t)i + 3tj<br /><br />4. Cartesian equation of the perpendicular bisector:<br />To convert the vector equation to the Cartesian equation, we can eliminate the parameter t:<br />(4+t)i + 3tj = (4i, 0)<br />4+t = 4<br />t = 0<br />0 = 3t<br />Therefore, the Cartesian equation of the perpendicular bisector is x = 4.<br /><br />ii) For points P(ai+bj) and Q(2ai+3bj):<br /><br />1. Midpoint of PQ:<br />Midpoint = ((ai+bj) + (2ai+3bj))/2 = (3ai+2bj)<br /><br />2. Direction vector of the perpendicular bisector:<br />The direction vector is perpendicular to PQ, so it is given by the cross product of the position vectors of P and Q:<br />Direction vector = (ai+bj) x (2ai+3bj) = (a, b) x (2a, 3b) = (3b-a, -2a+3b)<br /><br />3. Vector equation of the perpendicular bisector:<br />The vector equation of a line passing through the midpoint (3ai+2bj) and having the direction vector (3b-a, -2a+3b) is given by:<br />r = (3ai+2bj) + t(3b-a, -2a+3b) = (3ai+2bj+3bt-ataj, -2at+3bt)<br /><br />4. Cartesian equation of the perpendicular bisector:<br />To convert the vector equation to the Cartesian equation, we can eliminate the parameter t:<br />(3ai+2bj+3bt-ataj, -2at+3bt) = (3ai+2bj, 0)<br />3ai+2bj+3bt-ataj = 3ai+2bj<br />3bt-ataj = 0<br />t(3b-a) = 0<br />Therefore, the Cartesian equation of the perpendicular bisector is 3b-a = 0 or a = 3b.