Practice 1 Solutions

Practice 1 Solutions - Math E-21a Practice Exam #1...

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

View Full Document Right Arrow Icon
1 Math E-21a – Practice Exam #1 Solutions – Fall 2009 1) True/False questions (circle one) – you do not need to show your reasoning: a) The vector 1, 2, 4  v is perpendicular to the plane with equation 4 2 2 100 xyz  . False : In fact, the vector v is parallel to this plane. [It’s perpendicular to the normal vector 4,2,2 n .] b) If the vectors uv and are perpendicular, then the vectors u and v have the same length. True : Calculate 22 () 0    uv uv u v . Therefore . c) For any two vectors v and w , it is the case that 22 2  vw v w . False : 2 2  vw vw v uv w . This will only equal when either u0 , v0 , or u and v are perpendicular. d) For any three vectors u , v , w , the identity  uvw wvu holds. False :    uvw vwu wvu e) If 0  , then either 0 v or 0 w or . False : This will be the case whenever v and w are parallel. f) If u and v are perpendicular nonzero vectors, then uv u is parallel to v . True : Draw a picture. 2) Short Answer Questions – show your calculations and reasoning. a) Find the scalar projection (component) of the vector 5,1, 2 u in the direction of the vector 2,1,2 v . Solution : The scalar projection is given by 10 1 4 13 3 414     v u v . b) Find the vector projection of the vector u in the direction of the vector 2,1,2 v . Solution : The vector projection of u in the direction of v is 2 13 Proj 2,1,2 9      v vv u v uu v v . c) Find the angle between the vectors u and 2,1,2 v . Solution : For the angle between u and v, we use 1 13 cos 142.29 33 0  . d) Find the volume of the parallelepiped which contains the points (1,1,1) A , (2,3,4) B , (1 , 1 ,3 ) C , and (3,3,2) D , and where AB , AC , and AD are adjacent edges. Solution : The volume of the parallelepiped will be given by the (absolute value of the) triple scalar product: ( ) 1,2,3 ( 2,0,2 2,2,1 ) 1,2,3 4,6, 4 4 AB AC AD  , so the volume is 4.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/10/2011 for the course MATH E-21a taught by Professor - during the Fall '09 term at Harvard.

Page1 / 3

Practice 1 Solutions - Math E-21a Practice Exam #1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online