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1 Lernng Enhnement Tem Worsheet: The Cross Produt These re the model nswers for the worsheet tht hs questons on the ross produt etween vetors. The Cross Produt study gude. z x y. Loong t mge, you n see tht the vetors CD, CA nd CB n terms of the retngulr unt vetors, nd re: CD 0, CA 0 nd CB 0.
2 . The ross produt of two vetors nd n Dmensonl spe (D) s gven y:. For CD nd nd 0 : CA you hve tht,, 0,, CD CA ( ) ( ) 0 0 In the mge elow, you n see the vetor CD CA oloured purple. The result of the ross produt s vetor orthogonl (t rght-ngle) to oth CD nd CA. You n lso oserve tht the x nd y-oordntes of ths vetor re zero. z x y
3 . For CA nd nd 0 : CD you hve tht,, 0,, CA CD ( ) ( ) 0 0 In the mge elow, you n see the vetor CA CD oloured petrol lue. The result of the ross produt s vetor orthogonl (t rght-ngle) to oth CA nd CD. You n lso oserve tht the x nd y-oordntes of ths vetor re zero. z x y
4 . For CD nd 0 : CB you hve tht,, 0,, nd CD CB ( ) ( ) 0 0 In the mge elow, you n see the vetor CD CB oloured pn. The result of the ross produt s vetor orthogonl (t rght-ngle) to oth CD nd CB. You n lso oserve tht the x nd y-oordntes of ths vetor re zero z x y
5 v. For CB nd 0 : CD you hve tht,, 0,, nd CB CD ( ) ( ) 0 0 In the mge elow, you n see the vetor CB CD oloured rown. The result of the ross produt s vetors orthogonl (t rght-ngle) to oth CB nd CD. You n lso oserve tht the x nd y-oordntes of ths vetor re zero. z x y
6 . Loong t your nswers to prts () nd () you note tht: CD CA CA CD And loong t your nswers to prts () nd (v) you n see tht: CD CB CB CD So pee of mthemts whh desres ths s: For ny vetors nd :. The ross produt of two vetors nd n Dmensonl spe (D) s gven y:. For nd you hve tht,,,, nd. So, 8 8 Whh tells you tht the vetor 8 s norml (orthogonl) to oth nd. You n test whether two vetors re orthogonl y usng the dot produt s the dot produt of two orthogonl vetors s zero, see study gude: The Dot Produt. In other words heng tht the dot produts of the result wth oth nd re zero.
7 For : Also for : For nd, you hve tht, nd. So, 9,,, 8 Whh tells you tht the vetor 8 nd. s norml (orthogonl) to oth You n test whether two vetors re orthogonl y usng the dot produt s the dot produt of two orthogonl vetors s zero, see study gude: The Dot Produt. In other words heng tht the dot produts of the result wth oth nd re zero. For : Also for : For nd, you hve tht,,,, nd. So,
8 5 Whh tells you tht the vetor 5 s norml (orthogonl) to oth nd. You n test whether two vetors re orthogonl y usng the dot produt s the dot produt of two orthogonl vetors s zero, see study gude: The Dot Produt. In other words heng tht the dot produts of the result wth oth nd re zero. For : Also for : You re gven the followng vetors 5 nd.. In order to fnd vetor tht s norml (orthogonl) to oth vetors nd, you need to lulte ther ross produt:. The ross produt of two vetors nd n Dmensonl spe (D) s gven y:
9 You n use the ross produt to fnd the vetor tht s norml (orthogonl) to 5 nd. As, 5,,, nd : So, 0. In order to verfy tht your vetor s norml (orthogonl) to oth nd, you need to he tht the dot produts nd re zero. For : Also for : You n lulte the mgntude of usng the followng: n d.p. d. The ross produt of two vetors nd n the Dmensonl spe (D) s lso gven y the followng formul: sn nˆ. The mgntude of the ross produt s gven y: sn nˆ sn nˆ. The vetor nˆ s the unt vetor norml to oth nd, so n ˆ. And so: sn nˆ sn
10 To fnd the ngle etween the vetors nd you need to solve sn for. Rerrngng the equton sn for sn you hve sn. You now need to fnd. From prt of the queston, you hve tht 80.9 n d.p. You need to fnd the mgntude of the vetors nd. You n lulte the mgntude of usng the followng: 5 0 And the mgntude of 80 So, sn. Solvng the trgonometr equton you fnd tht 0 the ngle etween the two vetors s 90. You n he your nswer usng the dot produt... The two vetors nd re n the Dmensonl spe (D). So, ther ross produt s gven y: As,,,, 8 nd, ther ross produt s: The ross produt of the two vetors s the null vetor. Ths mens tht the vetors nd hve the sme dreton. You n lso oserve ths usng the followng formul:
11 sn nˆ The mgntude of the ross produt s gven y: sn nˆ sn nˆ. The vetor nˆ s the unt vetor norml to oth nd, so n ˆ. And so: sn nˆ sn To fnd the ngle etween the vetors nd you need to solve sn for. Rerrngng the equton sn for sn you hve sn. You hve found tht So, 0 0 nd sn 0. Solvng the trgonometr equton you fnd tht the ngle etween the two vetors s ether 0 or 80. Cheng the vetors nd you n see whether they hve the sme (ngle 0 ) or opposte (ngle 80 ) dreton nd dede whh of the ngles s the rght one. You n see tht, so the vetors hve the sme dreton nd 0.. To fnd the ngle etween the two vetors nd, you need to fnd the ross produt nd then usng the formul sn fnd the ngle etween them. The two vetors nd re n the Dmensonl spe (D). So, ther ross produt s gven y: As,,,, 8 nd, ther ross produt s: The ross produt of the two vetors s the null vetor. Ths mens tht the vetors nd hve the sme or opposte dreton. You n lso oserve ths usng the followng formul:
12 sn nˆ The mgntude of the ross produt s gven y: sn nˆ sn nˆ. The vetor nˆ s the unt vetor norml to oth nd, so n ˆ. And so: sn nˆ sn To fnd the ngle etween the vetors nd you need to solve sn for. Rerrngng the equton sn for sn you hve sn. You hve found tht So, 0 0 nd sn 0. Solvng the trgonometr equton you fnd tht the ngle etween the two vetors s ether 0 or 80. Cheng the vetors nd you n see whether they hve the sme (ngle 0) or opposte (ngle 80 ) dreton nd dede whh of the ngles s the rght one. You n see tht opposte dreton nd 80., so the vetors hve the 5.. Frst you need to fnd the ross produt As,,,, nd : Ther ross produt s: 0 Then lultng the ross produt you fnd tht:
13 0 Ths onfrms wht you oserved n queston. Tht: 0 0 So s n the opposte dreton of. To lulte wth, you frst need to lulte the ross produt. And now to lulte the dot produt : (to remnd yourself how to lulte the dot produt of two vetors see study gude: The Dot Produt) 8 To lulte, you frst need to lulte the ross produt. And now to lulte the dot produt : (to remnd yourself how to lulte the dot produt of two vetors see study gude: The Dot Produt)
14 Loong t your nswers you note tht: You n see tht for ny vetors, nd : The produts nd re lled slr trple produts. Ths worsheet s one of seres on mthemts produed y the Lernng Enhnement Tem wth fundng from the UEA Alumn Fund. Sn the QR-ode wth smrtphone pp for more resoures.
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