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# Lecture 3 MA 26100

Purdue

GPA 3.0

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This 3 page Class Notes was uploaded by Viktoryia Zhuleva on Tuesday January 19, 2016. The Class Notes belongs to MA 26100 at Purdue University taught by Patricia E Bauman in Fall 2015. Since its upload, it has received 53 views. For similar materials see Multivariable Calculus in Applied Mathematics at Purdue University.

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Date Created: 01/19/16

Lecture 3: Example: Find the point at which the line x=t, y=2-t, z=1+3t intersects the plane x+y+z=3. Solution: plug in x=t, y=2-t and z=1+3t into the plane equation x+y+z=3. t+(2-t)+(1+3t)=3 3t+3=3 3t=0 t=0 So the point is (x, y, z) = (0, 2, 1) Figure 1 3-dimentional triangle Definition: Two planes are called parallel id their normal vectors are parallel. (Non-parallel vectors intercept in a line). And the angle between the planes is equal to the angle between their normal vectors: ???? = cos1 ????1 2⃗.⃗⃗⃗⃗⃗ ????1|????2| Example: a) Find the angle between the planes: x + 2y + 2z = 1 and 3x – 4y + 10z = 8. Solution:⃗ =< ????,????,???? >=< 1,2,2 > ???????????? ???? ⃗ =< 3,−4,10 > 1 2 cos???? = (3 − 8 + 20) = 15 = √ 5 √1 + 4 + 4 √ + 16 + 100 √3 ∙ √ 5 5 5 ???? = cos −1√ 5 b) Find line of intersection (for x + 2y + 2z =1, 3x – 4y + 10z = 8) Solution: To find a point on this line try setting z=0. x + 2y = 1 Point (2, -1/2, 0) 3x – 4y = 8 3x + 6y = 3 3x – 4y = 8 10y = -5 y = -1/2, x = 2, z = 0 Example: Need a direction vector ???? for the line which is on both planes perpendicular to both planes. ???? ⊥⃗1,???? ⊥ ⃗2, which means that ???? = ⃗1× ????⃗2 ???? ???? ???? ???? = 1 2 2 =< 28,−4,−10 > 3 −4 10 ???? =< 14,−2,−5 > If there is a point P (2, -1/2, 0), the system equations are: ???? − 2 ???? + 1/2 2 = = 14 −2 −5 Cylinders and Quadratic Surfaces A cylinder is a surface that consists of all lines parallel to a given line that passes through a given plane curve. x +y =1 (z=0) Circular cylinder Figure 2 Cylinder in space Example: Perpendicularly stretch the graph of z=x (in space). Solution: notice that “y” doesn’t occur. Graph at y=0 is same shape as graph at y=k. Figure 3 Parabolic Cylinder Example: Graph y +4z =1 (Note: x is missing) Figure 4 Elliptic Cylinder

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