PROBLEMS 07 - VECTORS Page 1. Solve all problems vectorially: ( 1 ) Obtain the unit vectors perpendicular to each of , 29 , 29

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1 PROBLEMS 07 VECTORS Page Solve all problems vectorially: ( ) Obtain the unit vectors perpendicular to each x = ( ) d y = ( 0 ). ± ( ) If α is the gle between two unit vectors a d b then prove th l ā b α l = sin α. ( ) If a vector r makes Xaxis d Yaxis gles measures d 60 respectively then find the measure the gle which r makes Zaxis. [ Ans: 60 or 0 ] ( ) If x d y are noncollinear vectors R then prove th are noncoplar. x y d x y ( ) If the measure gle between x = i + j d y = ti j is π then find t. [ Ans: 0 ] ( 6 ) Show th for y a R the directions ( ) d ( a a + a + ) cnot be the same or opposite. ( 7 ) If θ is a measure gle between unit vectors a d b prove th sin θ = l a b l. ( 8 ) If x y d z are noncoplar prove th x + y y + z d z + x are also noncoplar.

2 PROBLEMS 07 VECTORS Page ( 9 ) Show th the vectors ( ) ( ) d ( ) are coplar. Also express each these vectors as a linear combinion the other two. ( ) = ( ) ( ) = ( ) + ( ); ( ) ( ) = ( ) ( ); [ Note: These vectors are collinear besides being coplar. Hence y vector R which is not collinear them cnot be expressed as a linear combinion these vectors even if it is coplar them. ] ( 0 ) Show th ( 0 ) ( 0 ) d ( 0 ) are noncoplar vectors. Also express y vector ( x y z ) R as a linear combinion these vectors. x + y z x y + z y + ( x y z ) = ( 0 ) + ( 0 ) + z x ( 0 ) ( ) Prove th gle in a semicircle is a right gle. ( ) Prove th the three altitudes in a trigle are concurrent. AP m ( ) If A P B d = then prove th for y point O in space PB n n ( OA ) + m ( OB ) = ( m + n ) OP. ( ) Prove th A ( 6 ) B ( ) d C ( 0 ) are collinear. Find also the rio in which A divides BC from B. [ Ans: : ] ( ) Find in which rio d which point does the XYple divide AB where A is ( ) d B is ( ). : from A 6 0

3 PROBLEMS 07 VECTORS Page ( 6 ) If A ( 0 ) B ( 6 ) d C ( 8 ) are given points then find the point D ( x y z ) in space so th AB = CD. [ Ans: ( 8 ) ] ( 7 ) A ( 0 ) B ( ) d C ( ) are given points. If D is the foot perpendicular from A on BC find its position vector. [ Ans: ( ) ] ( 8 ) If the position vectors A B C trigle ABC are a b c respectively then show th the area trigle ABC = ( a b ) + ( b c ) + ( c a ). ( 9 ) Find the volume a prism having a vertex origin O d having coterminous edges OA OB OC where A is ( ) B is ( ) d C is ( ). [ Ans: 0 cubic ] ( 0 ) Find the volume tetrahedron having vertices V ( ) A ( ) B ( 7 ) d C ( 6 8 ). cubic ( ) If the forces magnitudes d are applied to a particle in the directions vectors ( 0 ) ( 0 ) d ( ) respectively then find the magnitude d direction the resultt force. ( ) A bo is sailing to the a speed 0 km / hr. A m on bo feels th the wind is blowing from the south a speed km / hr. Find the true velocity the wind. km / hr gle 0 north

4 PROBLEMS 07 VECTORS Page ( ) A force magnitude 0 is acting on a particle in the direction i j d a force magnitude is acting on the same particle in the direction i + j. Under the influence these forces the particle is displaced from A ( ) to B ( 6 ). Find the work done. [ Ans: 7 ] ( ) Prove th the diagonals a rhombus bisect each other orthogonally. ( ) If a pair medis a trigle are equal then show th the trigle is isosceles. ( 6 ) Show th the perpendicular bisectors sides y trigle are concurrent. ( 7 ) Prove th the diagonals a rhombus are bisectors its gles. ( 8 ) If AD is a bisector BAC in trigle ABC d if D BC then show BD AB =. DC AC th ( 9 ) ABCDEF is a regular hexagon. Prove th AB + AC + AD + AE + AF = AD. ( 0 ) Show th centroid d incentre equileral trigle are the same. Find the incentre the trigle vertices ( 6 6 ) ( 0 ) d ( ). 0 ( ) If A is ( ) d B is ( ) then find S ( x y z ) such th AB = AS. [ Ans: ( 7 ) ]

5 PROBLEMS 07 VECTORS Page ( ) Let A ( ) d B ( ) be given points. Find in which rios from A d which points do the XY YZ d ZXples divide AB. : 0 ; : 0 ; AB is parallel to ZX ple ( ) Show th ( 6 0 ) ( 8 7 ) d ( 0 ) c be three vertices some rhombus. Find the coordines the fourth vertex this rhombus. [ Ans: ( 0 ) ] ( ) Show th ( ) ( ) ( 6 8 ) d ( ) are the vertices a trapezium. Find the area this trapezium. 9 ( ) Find the area the parallelogram ABCD if AC = a d BD = b. a b ( 6 ) Find the volume a prism having a vertex origin d having edges OA = i + j + k OB = i j + k d OC = i + j k. [ Ans: cubic ] ( 7 ) Show th ( ) ( 0 ) ( 9 ) d ( ) cnot be the vertices y tetrahedron. ( 8 ) Find the volume the tetrahedron vertices ( ) ( 0 ) ( 9 ) d ( ). 8 cubic

6 PROBLEMS 07 VECTORS Page 6 ( 9 ) A mechical bo is rowing the north speed 8 km / hr. If wind blows from the the speed 0 km / hr find the resulting speed the bo d also the direction resulting motion the bo. km / hr gle π north ( 0 ) A river flows a speed. A person desires to cross the river in a direction perpendicular to its flow. Find in which direction should he swim if his speed is 8. At gle π 8 the direction flow the river ( ) If speed a particle is the d 8 the southwest then find the resultt speed the particle d its direction. gle south ( ) A bo speeds the north 6. A m on the bo feels th the wind is blowing from the south. Find the true velocity the wind. 7 gle π north ( ) A steamer moves to the north a speed 0. A passenger on the steamer feels the wind to be blowing from the north. Find the true velocity the wind. gle 7 south ( ) A particle is displaced from A ( ) to B ( ) when forces magnitudes in the direction done. [ Ans: 0 ] i + j d 6 in the direction i j are applied. Find the work

[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]

[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989] VECTOR ALGEBRA o. Let a i be a vector which makes an angle of 0 with a unit vector b. Then the unit vector ( a b) is [MP PET 99]. The perimeter of the triangle whose vertices have the position vectors