1. 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.

2. 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

3. 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.
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4. 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:
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5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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.
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11. 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...
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12. 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.
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13. 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:
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14. 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.
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15. 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
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16. 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.

17. 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.
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