Vectors 5: The Vector Equation of a Plane Department of Mathematics University of Leicester For the second one, could you do something on how two lines intersect each other, and also where a line and a plane intersect.

Contents Introduction Getting to a point on a Line Vector Equation of a Plane Getting to a point on a Line Intersection of Lines with Planes

Introduction Vector Equation of a Plane Intersection of Lines with Planes Introduction We already know how to find the Vector Equation of a line, and we know about the Intersection of lines. Continuing from this, we will find the vector equation of planes, and also look at the intersection between a line and a plane. Next

Vector Equation of a Plane Introduction Vector Equation of a Plane Intersection of Lines with Planes Vector Equation of a Plane A plane is a flat 2D sheet in 3D space. eg: Next

Vector Equation of a Plane Introduction Vector Equation of a Plane Intersection of Lines with Planes Vector Equation of a Plane We use the same idea as for lines. This time, we need: A vector, a, to get TO the plane. 2 vectors, b and c, that are on the plane and are not parallel... a b c Next

Vector Equation of a Plane Introduction Vector Equation of a Plane Intersection of Lines with Planes Vector Equation of a Plane ... Then by choosing s and t, we can get to any point on the plane by doing . So the equation of the plane is (where s and t are variables) a b c Next

Vector Equation of a Plane Introduction Vector Equation of a Plane Intersection of Lines with Planes Vector Equation of a Plane What is the equation of the plane P joining , and ? Example: Answer: a = one point on P, b, c are vectors on P eg. So Next

Intersection of Lines with Planes Geometry Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Geometry The line is parallel to the plane There are 3 options: z The line intersects the plane at one place y x The line is on the plane Next

Intersection of Lines with Planes Geometry Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Geometry The line is parallel to the plane There are no intersection points The line intersects the plane at one place The intersection point is a single point The line is on the plane The intersection point is the line itself Next

Intersection of Lines with Planes Algebra Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Algebra We try to solve: are all fixed vectors. are the variables. We get 3 equations: etc. This is 3 simultaneous equations in 3 unknowns. Next

Intersection of Lines with Planes Algebra Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Algebra Eliminate t from the simultaneous equations. We then have 2 equations in and . Try to solve these... Next

Intersection of Lines with Planes Algebra Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Algebra No solution: The line is parallel to the plane Single solution: The line intersects the plane in one point Substitute and into the equation for P to find the intersection point. More than one solution: The line lies in the plane. Next

Intersection of Lines with Planes Example Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Example Where do L and P intersect? The equations are: We want to solve: Next

Intersection of Lines with Planes Example Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Example Eliminating t gives: So the only constraint on and is Whatever we choose for we can choose to make it work. So there are infinitely many solutions, and the points of intersection are the whole of L. Next

Intersection of Lines with Planes Question Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Question Where do L and P intersect? How many solutions do the simultaneous equations have? You should get Next

Intersection of Lines with Planes Question Introduction Vector Equation of a Plane Intersection of Lines with Planes Intersection of Lines with Planes Question Which of these lines is parallel to P? Hint: put then work out a.

Conclusion We can write Vector Equations for planes. Introduction Vector Equation of a Plane Intersection of Lines with Planes Conclusion We can write Vector Equations for planes. These make it easier to work out the points where a line intersects a plane (if there are any). Geometrically, there is more than one option for how a line and a plane intersect. Next