ajprincess
  • ajprincess
plssssssssssss help. (i)Expand (lxm).(lxn). Hence prove that the acute angle between two faces of a regular tetrahedron is cos^-1(1/3). (ii)Expand (lxm)x(lxn), and show its direction depends on whether the triad l,m,n is right- handed or left-handed. Hence prove that the volume of a regular tetrahedron is ((sqrt2)a^3)/12, where a is the length of each edge.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
aaha
anonymous
  • anonymous
this is vectors
anonymous
  • anonymous
gottit

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anonymous
  • anonymous
|dw:1333194259005:dw|
anonymous
  • anonymous
(i)Expand (lxm).(lxn). Hence prove that the acute angle between two faces of a regular tetrahedron is cos^-1(1/3) let \[\Pi _{1}, \Pi _{2}\] be the planes containing l,m and l,n respectively let alpha be the angle between the two planes, which we are trying to find (the cosine of) the acute angle between normals is equal to the angle between the planes, so: \[(l \times m).(l \times n) = (|l||m||\sin{\theta}|)(|l||n||\sin{\phi}|)\cos{\alpha}\] where theta and phi are angles between l and m , l and n respectively \[\text{in a regular tetrahedron we have equilateral triangles: } l=m=n \text{ , } \theta = \phi = 60\] hence:\[(l \times m).(l \times n) = (|l|^2\sin{60}|)(|l|^2\sin{60}|)\cos{\alpha}\] (more coming soon)
anonymous
  • anonymous
|dw:1333196596835:dw|
phi
  • phi
We could assign values to the vectors l,m,n and then do the indicated operation |dw:1333209375709:dw| The picture shows a tetrahedron with sides = 1. Using the vectors as defined, we get lxm= <0 0 sqrt(3)/2> normalized to unit length <0 0 1> and lxn= < 0 -sqrt(6)/3 sqrt(3)/6> with length 2/sqrt(3) lxn normalized to unit length is < 0 -2sqrt(2)/3 1/3> cos(A)= (lxn)/|lxn| dot (lxm)/|lxm| = 1/3 A= cos^-1 (1/3)
ajprincess
  • ajprincess
Thanks a lot for the help both of you.

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