Math 211: Worksheet 12.5 - Understanding Planes and Normal Vectors | Papers Advanced Calculus

Math 211: Worksheet 12.5 Day 6

Today you will be thinking about “shifting gears.” Imagine that this sheet of paper is very stiff

and extends off to infinity in all directions. So this is a plane, an R2, sitting inside of our three-

dimensional world, an R3. Now imagine that your pen is glued perpendicularly to the paper, this

is the “gear shift.” Any movement of the gear shift will also move the plane (the perpendicularity

must be preserved). Similarly, any movement of the plane will move the gear shift.

This little experiment in visualization shows us that if we start with a vector, the gear shift, there

is a unique plane perpendicular to the vector. Similarly, given a plane there is a unique direction

vector perpendicular to the plane (up to plus or minus). This perpendicular vector is called the

normal vector to the plane.

Let r=hx, y, zi. We will imagine ras the variable. Let n=ha, b, ci. The vector nwill always be

the normal vector.

(1) Calculate n·r.

(2) Every point perpendicular to nis on a plane with normal vector n. Since rgoes through the

origin setting the equation from (1) equal to zero gives the equation of the plane through

the origin with normal n. Write this equation down.

(3) What is the equation of the plane through the origin with normal vector h2,3,4i?

(4) What is the vector starting at h−5,2,4iand ending at r.

(5) If we want a plane through the point h−5,2,4iwith normal n, we set the dot product of n

with the answer from (3) equal to zero. Write this equation down as an equation of vectors

and dot products, then as an equation in coordinates.

(6) Replace the vector h−5,2,4iwith the vector r0=hx0, y0, z0iand write down the same

equations as you did in (4).

(7) Rewrite the equations from (5) with everything pertaining to ron one side of the equation

and everything pertaining to r0on the other.

(8) What is the equation of plane through (2,0,3) with normal vector h7,2,8iin coordinates?

(9) Any two vectors that are not on the same line determine a plane. What is a vector normal

to the plane containing vectors vand w?

(Hint: What vector is perpendicular to both vand w?)

(10) What is the coordinate equation of the plane through the origin determined by the vectors

h−2,3,4iand h1,0,8i?

(11) Given three points we can make two vectors, and from these two vectors we can make a

plane. Find the coordinate equation of the plane containing the three points h3,9,−2i,h4,1,0i,

and h1,2,−5i.

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Math 211: Worksheet 12.5 Day 6

Today you will be thinking about “shifting gears.” Imagine that this sheet of paper is very stiff and extends off to infinity in all directions. So this is a plane, an R^2 , sitting inside of our three- dimensional world, an R^3. Now imagine that your pen is glued perpendicularly to the paper, this is the “gear shift.” Any movement of the gear shift will also move the plane (the perpendicularity must be preserved). Similarly, any movement of the plane will move the gear shift.

This little experiment in visualization shows us that if we start with a vector, the gear shift, there is a unique plane perpendicular to the vector. Similarly, given a plane there is a unique direction vector perpendicular to the plane (up to plus or minus). This perpendicular vector is called the normal vector to the plane.

Let r = 〈x, y, z〉. We will imagine r as the variable. Let n = 〈a, b, c〉. The vector n will always be the normal vector.

(1) Calculate n · r. (2) Every point perpendicular to n is on a plane with normal vector n. Since r goes through the origin setting the equation from (1) equal to zero gives the equation of the plane through the origin with normal n. Write this equation down. (3) What is the equation of the plane through the origin with normal vector 〈 2 , 3 , 4 〉? (4) What is the vector starting at 〈− 5 , 2 , 4 〉 and ending at r. (5) If we want a plane through the point 〈− 5 , 2 , 4 〉 with normal n, we set the dot product of n with the answer from (3) equal to zero. Write this equation down as an equation of vectors and dot products, then as an equation in coordinates. (6) Replace the vector 〈− 5 , 2 , 4 〉 with the vector r 0 = 〈x 0 , y 0 , z 0 〉 and write down the same equations as you did in (4). (7) Rewrite the equations from (5) with everything pertaining to r on one side of the equation and everything pertaining to r 0 on the other. (8) What is the equation of plane through (2, 0 , 3) with normal vector 〈 7 , 2 , 8 〉 in coordinates? (9) Any two vectors that are not on the same line determine a plane. What is a vector normal to the plane containing vectors v and w? (Hint: What vector is perpendicular to both v and w?) (10) What is the coordinate equation of the plane through the origin determined by the vectors 〈− 2 , 3 , 4 〉 and 〈 1 , 0 , 8 〉? (11) Given three points we can make two vectors, and from these two vectors we can make a plane. Find the coordinate equation of the plane containing the three points 〈 3 , 9 , − 2 〉, 〈 4 , 1 , 0 〉, and 〈 1 , 2 , − 5 〉.

Math 211: Worksheet 12.5 - Understanding Planes and Normal Vectors, Papers of Advanced Calculus

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Math 211: Worksheet 12.5 Day 6