Question

18. Consider the line L with vector equation (x, y, z)-(3, 4,-1 1,-2, 5) and the point P(2, 5, 7). Show that P is not on L, and then find a Cartesian equation for the plane that contains both P and L.

Answer 1

"vectors eq^n of the line L: < x,y,z > = < 3,4,-1 > + t < 1, - 2,5 > and the point is given as p(2,5,7) from < 2,5,7 > = < 3,4,7 > + t < 1, - 2, 5 > we get, 2 = 3 + t, 5 = 4 - 2t , 7 = 7 + 5t which gives three different values of t hence, p(2,5,7) is not on the given line Now let us find the eq^n of the plane containing both p and L Now the points (2,6,7) and (3,4,7) are on the plane setting t = 1, we get another point on the plane as (4,2, 13) let a^vector be the vector from (4,2,12) to (2,5,7) and b^vector be the vector from (4,2,12) to (3,4,7) then a^vector = (- 2, 3,- 5) , b^vector = (- 1,2, - 5) let n^vector be normal of this two vector then , n^vector = a times b^vector =