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`ua n dv` are two non-collinear unit vectors such that `|( hat u+ hat v)/2+ hat uxx hat v|=1.` Prove that `| hat uxx hat v|=|( hat u- hat v)/2|dot`

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Given that `|(hatu+hatv)/2+hatuxxhatv|=1`
` |(hatu+hatv)/2+hatuxxhatv|^(2)=1`
`(2+2costheta)/4+sin^(2)theta=1`
`cos^(2)(theta /2)=cos^(2)theta`
`theta=npi+-theta/2,n inZ`
`(2pi)/3`
`|hatuxxhatv|=sin((2pi)/3)=sin (pi/3)=|(hatu-hatv)/2|`
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Write a unit vector perpendicular to hat i+ hat j\ a n d\ hat j+ hat kdot

The deges of a parallelopied are of unit length and are parallel to non- coplanar unit vector hat(a), hat(b) , hat(c ) such that hat(a) , hat(b) = hat(b), hat( c)=hat(c ), hat(a) = .(1)/(2). Then the volume of the parallelopiped is

Let hat u= hat i+ hat j , hat v= hat i- hat ja n d hat w= hat i+2 hat j+3 hat kdot If hat n is a unit vector such that hat udot hat n=0a n d hat vdot hat n=0, then find the value of | hat wdot hat n|dot

Let two non-collinear unit vector hat a a n d hat b form an acute angle. A point P moves so that at any time t , the position vector O P(w h e r eO is the origin) is given by hat acost+ hat bsintdotW h e nP is farthest from origin O , let M be the length of O Pa n d hat u be the unit vector along O Pdot Then (a) hat u=( hat a+ hat b)/(| hat a+ hat b|)a n dM=(1+ hat adot hat b)^(1//2) (b) hat u=( hat a- hat b)/(| hat a- hat b|)a n dM=(1+ hat adot hat )^(1//2) (c) hat u=( hat a+ hat b)/(| hat a+ hat b|)a n dM=(1+2 hat adot hat b)^(1//2) (d) hat u=( hat a- hat b)/(| hat a- hat b|)a n dM=(1+2 hat adot hat b)^(1//2)

If vec a and vec b are non-zero and non-collinear vectors, then vec ax vec b=[ vec a vec b hat i] hat i+[ vec a vec b hat j] hat j+[ vec a vec b hat k] hat k vec adot vec b=( vec adot vec i)( vec adot hat i)( vec bdot hat j)+( vec adot hat j) ( vec bdot hat j) + ( vec adot hat k)( vec bdot hat k) If vec u= hat a-( hat adot hat b) hat b and hat v= hat ax hat b , then | vec v|=| vec u| If vec c= vec ax( vec ax vec b) , then vec cdot vec a=0

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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
  1. ua n dv are two non-collinear unit vectors such that |( hat u+ hat ...

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  2. Find |veca| and |vecb| if (veca+vecb).(veca-vecb)=8 and |veca|=8|vecb|...

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  3. Show that |veca|vecb+|vecb|veca is perpendicular to |veca|vecb-|vecb|v...

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  4. If the vectors A ,B ,C of a triangle A B C are (1,2,3),(-1,0,0),(0,1,2...

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  5. If |veca|=3, |vecb|=4 and the angle between veca and vecb is 120^(@) ....

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  6. If vectors hati-2xhatj-3yhatk and hati+ 3 x hatj + 2yhatk are orthogon...

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  7. Let veca vecb and vecc be pairwise mutually perpendicular vectors, suc...

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  8. If veca +vecb +vecc=0, |veca|=3,|vecb|=5, |vecc|=7 , then find the ang...

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  9. If the angle between unit vectors veca and vecb is 60^(@) . Then find ...

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  10. Let vecu=hai+hatj,vecv=hati-hatj and vecw=hati+2hatj+3hatk. If hatn is...

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  11. A , B , C , D are any four points, prove that vec A Bdot vec C D+ ...

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  12. P(1,0,-1), Q(2,0,-3),R(-1,2,0)a n dS(,-2,-1), then find the projection...

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  13. If the vectors 3vecP+vecq, 5vecP - 3vecq and 2 vecp+ vecq, 4 vecp - 2v...

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  14. Let vecA and vecB be two non-parallel unit vectors in a plane. If (alp...

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  15. Let veca , vecb and vecc be unit vectors such that veca+vecb+vecc=vecx...

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  16. If veca and vecb are unit vectors, then find the greatest value of |ve...

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  17. Constant forces P(1)=hati-hatj+hatk, P(2)= -hati+2hatj-hatik and P(3)=...

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  18. Find |veca| and |vecb| if (veca+vecb).(veca-vecb)=8 and |veca|=8|vecb|...

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  19. If A ,B ,C ,D are four distinct point in space such that A B is not...

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  20. If veca=2hati+ 3hatj-5hatk,vecb =m hati+nhatj +12 hatk and veca xx vec...

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  21. Find veca.vecb if |veca|2, |vecb|=5,a and |vecaxxvecb|=8

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