A-Level Vector geometry in three dimensions: worked solution

Question

With l: r = (2, 1, 0) + lambda(3, -2, 1) and rooftop p: 2x + y - z = 7, on Day 2 the drone descends from A(2, 1, 0) to the rooftop by the shortest route. Find the exact landing coordinates.

Step 1: decide which direction the drone travels. The shortest route from a point to a plane is along the plane's normal (any other route is the hypotenuse of a right-angled triangle whose shorter side is the perpendicular). For the plane 2x + y - z = 7 the coefficients give the normal vector n = (2, 1, -1). Note the line l is NOT the route here: "shortest route to the rooftop" means perpendicular to the rooftop, so its direction (3, -2, 1) is not used. Step 2: write the descent path as a line through A(2, 1, 0) in the direction of n. r = (2, 1, 0) + mu(2, 1, -1) A general point on this path is (2 + 2mu, 1 + mu, 0 - mu) = (2 + 2mu, 1 + mu, -mu). Step 3: find where the path meets the plane by substituting the general point into 2x + y - z = 7. 2(2 + 2mu) + (1 + mu) - (-mu) = 7 4 + 4mu + 1 + mu + mu = 7 5 + 6mu = 7 6mu = 2 mu = 1/3 Step 4: substitute mu = 1/3 back into the general point to get the landing coordinates. x = 2 + 2(1/3) = 2 + 2/3 = 8/3 y = 1 + 1/3 = 4/3 z = -1/3 Landing point: (8/3, 4/3, -1/3) Check it lies on the plane: 2(8/3) + 4/3 - (-1/3) = 16/3 + 4/3 + 1/3 = 21/3 = 7. Correct.