# Vector Analysis Sample Essay

1. 1. 1 Show how to happen A and B. given A+B and A ?B.
1. 1. 2 The vector A whose magnitude is 1. 732 units makes equal angles with the co-ordinate axes. Find Ax. Ay. and Az. 1. 1. 3 Calculate the constituents of a unit vector that lies in the xy-plane and makes equal angles with the positive waies of the x- and y-axes. 1. 1. 4 The speed of sailing boat A relation to sailboat B. vrel. is defined by the equation vrel = Virginia ? vB. where Virginia is the speed of A and vB is the speed of B. Determine the speed of A relation to B if vA = 30 km/hr east

vB = 40 km/hr North.
ANS. vrel = 50 km/hr. 53. 1? South of E.
1. 1. 5 A sailing boat canvas for 1 hour at 4 km/hr ( comparative to the H2O ) on a steady compass header of 40? E of North. The sailing boat is at the same time carried along by a current. At the terminal of the hr the boat is 6. 12 kilometer from its get downing point. The line from its get downing point to its location lies 60? E of north. Find the x ( easterly ) and y ( northwards ) constituents of the water’s speed. ANS. veast = 2. 73 km/hr. vnorth ? 0 km/hr.

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1. 1. 6 A vector equation can be reduced to the signifier A = B. From this show that the one vector equation is tantamount to three scalar equations. Assuming the cogency of Newton’s 2nd jurisprudence. F = mom. as a vector equation. this means that ax depends merely on Fx and is independent of Fy and Fz. 1. 1. 7 The vertices A. B. and C of a trigon are given by the points ( ?1. 0. 2 ) . ( 0. 1. 0 ) . and ( 1. ?1. 0 ) . severally. Find point D so that the figure ABCD forms a plane parallelogram. ANS. ( 0. ?2. 2 ) or ( 2. 0. ?2 ) . 1. 1. 8 A trigon is defined by the vertices of three vectors A. B and C that extend from the beginning. In footings of A. B. and C show that the vector amount of the consecutive sides of the trigon ( AB +BC +CA ) is nothing. where the side AB is from A to B. etc. 1. 1. 9 A domain of radius a is centered at a point r1. ( a ) Write out the algebraic equation for the domain.

( B ) Write out a vector equation for the domain.
ANS. ( a ) ( ten ?x1 ) 2 + ( y ?y1 ) 2 + ( z ?z1 ) 2 = a2.
( B ) R = r1 +a. with r1 = centre.
( a takes on all waies but has a fixed magnitude a. )
1. 1. 10 A corner reflector is formed by three reciprocally perpendicular reflecting surfaces. Show that a beam of light incident upon the corner reflector ( striking all three surfaces ) is reflected back along a line analogue to the line of incidence.

Hint. See the consequence of a contemplation on the constituents of a vector depicting the way of the light beam.
1. 1. 11 Hubble’s jurisprudence. Hubble found that distant galaxies are withdrawing with a speed proportional to their distance from where we are on Earth. For the ith galaxy. six = H0ri. with us at the beginning. Show that this recession of the galaxies from us does non connote that we are at the centre of the existence. Specifically. take the galaxy at r1 as a new beginning and show that Hubble’s jurisprudence is still obeyed.

1. 1. 12 Find the diagonal vectors of a unit regular hexahedron with one corner at the beginning and its three sides Liing along Cartesian co-ordinates axes. Show that there are four diagonals with length [ movie ] . Representing these as vectors. what are their constituents? Show that the diagonals of the cube’s faces have length v2 and find their constituents.

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