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    The question said "whats the shortest distance between A and the line OB?"

     A = i +5j -k , B= 6i +3j -6k.
    So line OB is  \lambda (3i +j -3k)
    Now I tryed to find the distance using pythag.
     x^2 = \lambda ^2(3^2+1^2+3^2) + (1^2+5^2+1^2)= \lambda ^2(19) + 27
    surely, because any number squared is positive, the shortest distance is when lambda equals 0. so i got
     x = \sqrt{27}
    but the mark scheme did
     distance = OA \sin{\theta} = 3 \sqrt{3} * \sqrt{\frac{2}{3}} = 3 \sqrt{2} (we had to work out sin(theta) earlier)
    I'm strugling to picture the 3 dimensions in my head. how does the sin work? and why doesn't my method work?
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    (Original post by hello calum)
    The question said "whats the shortest distance between A and the line OB?"

     A = i +5j -k , B= 6i +3j -6k.
    So line OB is  \lambda (3i +j -3k) <-------
    Now I tryed to find the distance using pythag.
     x^2 = \lambda ^2(3^2+1^2+3^2) + (1^2+5^2+1^2)= \lambda ^2(19) + 27
    surely, because any number squared is positive, the shortest distance is when lambda equals 0. so i got
     x = \sqrt{27}
    but the mark scheme did
     distance = OA \sin{\theta} = 3 \sqrt{3} * \sqrt{\frac{2}{3}} = 3 \sqrt{2} (we had to work out sin(theta) earlier)
    I'm strugling to picture the 3 dimensions in my head. how does the sin work? and why doesn't my method work?
    Just there.
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    The direction from A to a general point on OB is (i+5j-k)-L(2i+j-2k).

    The dot product of this with 2i+j-2k needs to be zero.
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    Remember that the shortest distance is always a perpendicular line.
 
 
 
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