notes001 (42) - t h& 1& 2 i Parametric equations for...

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September 3, 2010 NAME_______________________________ 1) Let u = 3 i + 3 j + 3 k and v = 3 i 2 j k . Compute u ± v (using any Solution: u ± v = i j k 3 3 3 3 2 1 = 3 3 2 1 i 3 3 3 1 j + 3 3 3 2 k = 3 i + 6 j 3 k . 2) Show that u ± v (which you computed in problem 1) is orthogonal to both u and v . Solution: ( u ± v ) ² u = (3 i + 6 j 3 k ) ² ( 3 i + 3 j + 3 k ) = (3) ( 3) + (6) (3) + ( 3) (3) = 0 and ( u ± v ) ² v = (3 i + 6 j 3 k ) ² (3 i 2 j k ) = (3) (3) + (6) ( 2) + ( 3) ( 1) = 0 so u ± v is orthogonal to both u and v . 3) Find either parametric equations or symmetric equations for the line that contains the points P 0 ( 4 ; 5 ; 4) and P 1 ( 5 ; 3 ; 4) . Solution: The line in question contains the point P 0 and is parallel to the vector v = P 0 P 1 = 5 ( 4) ; 3 5 ; 4 ( 4) i = 1 ; 2 ; 0 i . 1
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Thus a vector equation for the line is OP = OP 0 + t v or h x; y; z i = 4 ; 5 ; 4 i
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Unformatted text preview: + t h& 1 ; & 2 ; i . Parametric equations for this line are x = & 4 & t y = 5 & 2 t z = & 4 &1 < t < 1 . Symmetric equations are x & 4 & 1 = y & 5 & 2 z = & 4 . 4) Find either parametric equations or a component&form equation for the plane that contains the point P (5 ; 1 ; & 4) and is perpendicular to the vector n = h& 1 ; & 1 ; & 5 i . Solution: A component&form equation for this plane is n ± &&! P P = 0 or h& 1 ; & 1 ; & 5 i ± h x & 5 ; y & 1 ; z + 4 i = 0 or & ( x & 5) & ( y & 1) & 5 ( z + 4) = 0 or & x + 5 & y + 1 & 5 z & 20 = 0 or & x & y & 5 z = 14 . 2...
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