{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

102f05fin

# 102f05fin - B U Department of Mathematics Math 102 Calculus...

This preview shows pages 1–2. Sign up to view the full content.

B U Department of Mathematics Math 102 Calculus II Fall 2005 Final This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. (a) Find an equation for the plane through A (0 , 0 , 1) , B (2 , 0 , 0) , and C (0 , 3 , 0). (b) Find the cosine of the angle between this plane and xy-plane. (c) Find the area of the triangle ABC. Solution: (a) We find a vector normal to the plane and use it with one of the points (it doesn’t matter which) to write an equation for the plane. -→ AB = < 2 - 0 , 0 - 0 , 0 - 1 > = < 2 , 0 , - 1 > -→ AC = < 0 - 0 , 3 - 0 , 0 - 1 > = < 0 , 3 , - 1 > The cross product -→ AB × -→ AC = i j k 2 0 - 1 0 3 - 1 = 3 i + 2 j + 6 k is normal to the plane. By using components of this vector and coordinates of the point (0,0,1) we can write the equation of the plane as 3( x - 0) + 2( y - 0) + 6( z - 1) = 0 3 x + 3 y + 6 z = 6 (b) The angle between two intersecting planes is defined to be the (acute) angle determined by their normal vectors. The vectors n 1 = < 0 , 0 , 1 > and n 2 = < 3 , 2 , 6 > are normal vectors for xy-plane and the plane 3 x + 2 y + 6 z = 6 , respectively. To find the cosine of the angle

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern