Begin2.DVI

I6-28. (a) If ˆe1 = cosα 1 ˆe 1 + cosβ 1 ˆe 2 + cosγ 1 ˆe 3 and ˆe 2 = cosα 2 ˆe 1 + cosβ 2 ˆe 2 + cosγ 2 ˆe 3 ,

then

ˆe` 1 ·ˆe` 2 = cosθ= cosα 1 cosα 2 + cosβ 1 cosβ 2 + cosγ 1 cosγ 2

(b)θ= cos−^1 (8/9) =. 475882 radians≈ 27. 266 ◦

I6-29. Shortest distance is 9 units.

I6-31. IfA~×B~ =~ 0 , thenA~is colinear withB~ and ifB~×C~=~ 0 , thenB~ is colinear

withC~. Therefore,A~is colinear withC~so thatA~×C~=~ 0.

I6-32. Normal to plane isN~= (~r 3 −~r 1 )×(~r 2 −~r 1 )

Equation of plane is(~r−~r 1 )·N~= 0or

(x−3)−2(y−10) + 2(z−13) = 0The distance from

given point to plane is projection of(~r 0 −~r 1 )onto

ˆeNgiving 9 units for the distance.

Here~r 0 = 6ˆe 1 + 3ˆe 2 + 18ˆe 3

I6-33.

M~P=~r 1 ×F~

~r 1 =~r+A~

M~P=(~r+A~)×F~=~r×F~+A~×F~=~r×F~

becauseA~×F~=~ 0 , A~andF~ are colinear

M~P·ˆeL=projection ofM~Pon the lineL

I6-34. (a) M~ 0 =~r 1 ×F~=− 1200 ˆe 1 +800ˆe 2 +200ˆe 3

(b) M~P 2 = (~r 1 −~r 2 )×F~=− 1400 ˆe 1 +1400ˆe 2 +700ˆe 3

(c) ˆeL=−ˆe^1 + 3√ˆe^2 −^4 ˆe^3

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ML=M~P 2 ·ˆeL=^2800 √

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Solutions Chapter 6