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Vectors 5: The Vector Equation of a Plane Department of Mathematics University of Leicester

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

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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. Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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

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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... Vector Equation of a Plane a b c Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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... 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) Vector Equation of a Plane a b c Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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

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Intersection of Lines with Planes Geometry There are 3 options: 1.The line is parallel to the plane 2.The line intersects the plane at one place 3.The line is on the plane x y z Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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Intersection of Lines with Planes Geometry 1.The line is parallel to the plane There are no intersection points 2.The line intersects the plane at one place The intersection point is a single point 3.The line is on the plane The intersection point is the line itself Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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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. Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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

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Intersection of Lines with Planes Algebra 1.No solution: The line is parallel to the plane 2.Single solution: The line intersects the plane in one point Substitute and into the equation for P to find the intersection point. 3.More than one solution: The line lies in the plane. Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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

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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. Intersection of Lines with Planes Example Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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

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

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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. Conclusion Introduction Intersection of Lines with Planes Vector Equation of a Plane Next

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