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07%20-%20Vectors - PROBLEMS 07 VECTORS Page 1 Solve all...

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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 X-axis and Y-axis angles of measures 45 ° and 60 ° respectively, then find the measure of the angle which - r makes with Z-axis. [ Ans: 60 ° or 120 ° ] ( 4 ) If - x and - y are non-collinear vectors of R 3 , then prove that - x , - y and - - y x × are non-coplanar. ( 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 non-coplanar, prove that y x + , z y + and x z + are also non-coplanar.
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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 non-coplanar vectors. Also express any vector ( x, y, z ) of R 3 as a linear combination of these vectors. + + + + + = ) 1 1, 0, ( 2 x z y ) 1 0, 1, ( 2 z y x ) 0 1, 1, ( 2 z y x ) z y, x, ( : Ans - - - ( 11 ) Prove that an angle in a semi-circle is a right angle.
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