Vector Algebr nd Clculus I (23) 2AA 2AA Vector Algebr nd Clculus I Bugs/queries to sjrob@robots.ox.c.uk Michelms 23. The tetrhedron in the figure hs vertices A, B, C, D t positions, b, c, d, respectively. () Derive n expression for the unit norml ˆn bc to the plnr fce ABC. Explin why points r on the plne ABC cn be written r ˆn bc = ˆn bc. Hence, nd by writing y = d+βˆn bc, derive the position of Y, the projection of D onto the fce ABC long the norml to ABC. (b) By writing x = + α(c ), derive n expression for the position vector x of the point X on the line AC such tht DX is perpendiculr to AC. (c) The unit norml you derived in prt () might point into, or out of, the tetrhedron. With knowledge of the ordering of vertices round ech fce (sy from the digrm) it is of course possible to recover n outwrd fcing norml. Now suppose tht you do not know the ordering. (i) By considering the vector from D to A, devise procedure to check whether the ˆn bc you recovered in prt () is unit norml pointing out of the tetrhedron. (ii) Would using the wrong sign for norml ffect the result of prt ()? (d) In terms of outwrds fcing unit normls, derive n expression for the internl ngle between fces ABC nd BCD. Would using the wrong sign for norml ffect the result of prt (d)? D B O A x X Y C
2 Vector Algebr nd Clculus I (23) 2AA 2. The figure shows lser mounted on pn-tilt mechnism so tht its bem cn be directed from the point in ny direction ˆb. The bem must be reflected from plne mirror with norml ˆn fcing out of the mirror to hit trget lying t position t. () Derive the eqution of the mirror plne. (b) Derive n expression for the minimum stright-line distnce of point t from the plne mirror. (c) Show tht the direction the lser must point in is ˆb = v/ v where v = t 2 ((t m) ˆn) ˆn. (d) A vlid trget position t must lie in front of the mirror. Devise vector test for this cse. n 0 Plne Mirror 0 0 Trget t b Lser m O
Vector Algebr nd Clculus I (23) 2AA 3 3. The figure shows two ground-bsed rdr sttions trcking n ircrft. Sttion hs fixed Crtesin coordinte system (Oxyz) ttched to it, nd mesures the direction to n ircrft s ˆn, but it cnnot determine the rnge ρ. Sttion 2 is locted t in the first s coordinte system. In its locl coordinte system (Oxyz) 2 it simultneously mesures the direction to the ircrft s ˆn 2 but gin does not determine the rnge ρ 2. () Assume tht the two locl coordinte systems re ligned (ie there is no rottion between them). Show tht the ircrft s position in (Oxyz) cn be written in two wys, s either p = ρ ˆn or s p = + ρ 2ˆn 2. Hence show tht the rnge from the first sttion is ρ = ( ˆn 2) (ˆn ˆn 2 ) ˆn ˆn 2 2 (b) The ircrft is t p = [ 20, 5, 5] km reltive to (Oxyz), nd = [, 0, 0.3] km. Use vector mgnitude to derive the rnge ρ, nd deduce ˆn. Also derive the position p 2 of the ircrft s mesured in (Oxyz) 2, nd hence find ˆn 2. (c) Verify the formul derived in prt () by inserting the vlues of ˆn,2 found in prt (b). (d) Wht simple chnge to the expression for ρ would you hve to mke if the two coordinte systems were still seprted by but were no longer ligned? (e) Under wht circumstnces would the expression of prt () brek down? Relte your nswers to the physicl sitution. ρ 2 n 2 Sttion 2 ρ n Sttion
4 Vector Algebr nd Clculus I (23) 2AA 4. () The vector triple-product is p (q r) = (p r)q (p q)r. Strting from the cyclic property of the sclr triple-product (p q) r = (r p) q, use judicious substitution of vectors to i. Express ( b) (c d) in terms of sclr products only; nd hence ii. Show tht ( b) (( c) d) = ( d)( (b c)). (b) i. Express ( b) (c d) s liner combintion of c nd d. ii. By writing the expression in squre brckets s liner combintion of nd c, simplify ( b) [(b c) (c )]. (c) By writing x = λ + µb + ν( b) (where nd b re non-prllel vectors, nd where λ, µ nd ν re sclrs), derive generl solutions to the following equtions i. x = b, given tht b = 0. ii. x = γ. 5. A firground ride comprises disc in the horizontl xy-plne, its centre t the origin of the world coordinte system. A trck runs from the centre to the edge of the disk nd is initilly coincident with the world s î xis. At time t = 0 (i) cr moves off long the trck from the centre of the disk with constnt speed ν, nd (ii) the disk is set to rotte with ngulr velocity ωˆk bout its centre. () Drw digrm nd confirm tht in the world coordinte system the cr s position t time t is R(t) = νtˆr(t), nd deduce n expression for ˆr(t). (b) Show tht the cr s velocity in the world frme is dr/dt = νˆr(t)+νtωˆθ(t). (c) Derive the cr s ccelertion in the world frme, nd interpret the terms you find. (d) Repet steps () - (c) for the cse where the cr moves off long the trck with constnt ccelertion α. Agin discuss the terms present in the cr s ccelertion.
Vector Algebr nd Clculus I (23) 2AA 5 6. The mutully perpendiculr unit vectors ˆl, ˆm, ˆn re functions of time t, but remin mutully perpendiculr unit vectors. As ˆl, ˆm, ˆn form bsis, it must be the cse tht dˆl/dt = α ˆl + β ˆm + γ ˆn, where α etc re (s yet) rbitrry coefficients; nd similrly for d ˆm/dt nd dˆn/dt. () By differentting ˆl ˆl = w.r.t. time, nd similrly for ˆm nd ˆn, show tht dˆl/dt = β ˆm + γ ˆn d ˆm/dt = α 2ˆl + γ 2ˆn dˆn/dt = α 3ˆl + β 3 ˆm (b) By differentiting ˆl ˆm = 0, nd similrly for the other vectors, obtin three reltionships of the form α 2 + β = 0. Hence, nd using the vector test for coplnrity, show tht dˆl/dt, d ˆm/dt, nd dˆn/dt, re coplnr. (c) Consider physiclly how three mutully perpendiculr unit direction vectors cn chnge, nd so justify this coplnrity result. (d) Confirm tht your expressions obtined erlier re verified when you differentite the expression ˆl ˆm = ˆn with respect to t. 7. The xis of helicl tube with cross-sectionl re A is given by r(p) = [ sin p, cos p, bp]. The tube crries n inviscid, incompressible fluid t volume flow rte of L units per second. () Find unit tngent vector ˆt to the xis, nd hence determine p in terms of rc-length, s. (b) By determining the reltionship between rc-length s nd time t, write down the instntneous velocity vector of n element of fluid on the xis. (c) Find dˆt/ds nd the curvture of the xis. Hence determine the direction nd mgnitude of the instntneous ccelertion of n element of fluid on the xis. Discuss the signs of the components of this ccelertion vector.
6 Vector Algebr nd Clculus I (23) 2AA 8. () A sclr potentil field is given by φ nd the vector field F is F = î φ x + ĵ φ y + ˆk φ z. Noting tht dr = îdx + ĵdy + ˆkdz, nd reclling the definition of perfect or totl differentil, show tht the line integrl B A F dr = B A dφ = φ B φ A. (b) The vector field F = yzî + xzĵ + xyˆk. Evlute the line integrl B A F dr, where A is (0, 0, 0) nd B is (4, 2, 4), in two wys: i. by integrting to find tht potentil function φ exists, nd evluting the difference in potentil between the points. ii. by integrtion long the prmetrized curve x = p 2, y = p, z = 2p. To do this, express F in terms of p, nd dr in terms of p nd dp. 9. () Write down vector expressions for the elements of line dr nd surfce ds nd sclr expressions for the elements of volume dv shown in the figure, using the pproprite coordinte system. (b) Find the work done F dr in field F(x, y) = yî + xĵ when moving long i. the shorter circulr pth from (x, y) = (, 0) to (x, y) = (/ 2, / 2); ii. the longer pth from (x, y) = (/ 2, / 2) to (x, y) = (, 0). (c) Find S v ds where v = r 2 (xî + yĵ + zˆk) nd S is the entire surfce of the sphere x 2 + y 2 + z 2 = 2. y dr x ds z z ds dv dv ds
Vector Algebr nd Clculus I (23) 2AA 7 Answers nd Hints. () y = d + (( d) ˆn bc )ˆn bc (b) α = (d ) (c )/ c 2 2. (b) α = (t m) ˆn (c) Think where the imge of t ppers... 3. () Think bout vector opertion to eliminte ρ 2 from the eqution. (b) Using p lone, ρ = 450 km ˆn = [ 0.942, 0.236, 0.236], ˆn 2 = [ 0.9505, 0.2263, 0.227] (c) Intermedite checks... ˆn ˆn 2 = [0.035, 0.0235, 0.4374], ˆn 2 = [2.953, 0.4979, 9.279]. 4. () ( c)(b d) ( d)(b c) then use replcement to continue. (b) [ (b c)] 2 or cyclic permuttions thereof. (c) x = λ + µb + γ b, where (i) λ rbitrry, µ = 0, nd ν = / 2 (ii) λ rbitrry, µ = (γ λ 2 )/ b, nd ν rbitrry. 5. (d) R = α( t 2 ω 2 /2)ˆr + 2αtωˆθ 6. (d) Hint: differentite both sides, nd then use prts () nd (b) to show tht they re the sme. 7. () ˆt(p) = ( 2 + b 2 ) /2 [ cos p, sin p, b], p = ( 2 + b 2 ) /2 s; (b) ds/dt = L/A, v(p) = L/Aˆt(p); (c) κ = /( 2 + b 2 ), ˆn = [ sin p, cos p, 0], dv/dt = (L 2 /A 2 )κˆn. 8. (b) i) 32 nd (ii) 32 9. () Consult the lecture notes (b) 2 /2 nd 2 /2. Think why they sum to zero. (c) 4π 5