This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: PROBLEMS 07  VECTORS Page 1 Solve all problems vectorially: ( 1 ) Obtain the unit vectors perpendicular to each of  x = ( 1, 2,  1 ) and  y = ( 1, 0, 2 ). ± 29 2 , 29 3 , 29 4 : Ans ( 2 ) If α is the angle between two unit vectors  a and  b , then prove that l  a   b cos α l = sin α . ( 3 ) If a vector  r makes with Xaxis and Yaxis angles of measures 45 ° and 60 ° respectively, then find the measure of the angle which  r makes with Zaxis. [ Ans: 60 ° or 120 ° ] ( 4 ) If  x and  y are noncollinear vectors of R 3 , then prove that  x ,  y and  y x × are noncoplanar. ( 5 ) If the measure of angle between x = j i + and y = t i  j is 4 3 π , then find t. [ Ans: 0 ] ( 6 ) Show that for any a ∈ R, the directions ( 2, 3, 5 ) and ( a, a + 1, a + 2 ) cannot be the same or opposite. ( 7 ) If θ is a measure of angle between unit vectors a and b , prove that sin 2 θ = 2 1 l a  b l . ( 8 ) If z and y , x are noncoplanar, prove that y x + , z y + and x z + are also noncoplanar. PROBLEMS 07  VECTORS Page 2 ( 9 ) Show that the vectors ( 1, 2, 1 ), ( 1, 1, 4 ) and ( 1, 3,  2 ) are coplanar. Also express each of these vectors as a linear combination of the other two. = = + = ) 4 1, 1, ( ) 1 2, 1, ( 2 ) 2 3, 1, ( ); 2 3, 1, ( ) 1 2, 1, ( 2 ) 4 1, 1, ( ); 2 3, 1, ( 2 1 ) 4 1, 1, ( 2 1 ) 1 2, 1, ( : Ans [ Note: These vectors are collinear besides being coplanar. Hence, any vector of R 3 which is not collinear with them cannot be expressed as a linear combination of these vectors even if it is coplanar with them. ] ( 10 ) Show that ( 1, 1, 0 ), ( 1, 0, 1 ) and ( 0, 1, 1 ) are noncoplanar vectors. Also express any vector ( x, y, z ) of R 3 as a linear combination of these vectors....
View
Full
Document
This note was uploaded on 11/28/2011 for the course PHYSICS 300 taught by Professor Smith during the Spring '06 term at ITT Tech Flint.
 Spring '06
 Smith

Click to edit the document details