I need to find the equation of a plane that passes through the point with position vector A = 5 i + j + 3 k and the line l 1 = 2 i − 8 j − 17 k + λ ( i − 3 j − 4 k )I found the normal by finding the cross product of A ∗ [ ( 5 i + j + 3 k ) − ( 2 i − 8 j − 17 k ) ] which turned out to be n 1 = − 24 i − 32 j + 18 k. Then I used r ⋅ n = a ⋅ n → ( 5 i + j + 3 k ) ⋅ ( − 24 i − 32 j + 18 k ) = − 98However, it turns out that the above value should be 49 rather than −98. I realize that it can be divided by -2 to get the answer; however, I don't understand why this happens.

Question

I need to find the equation of a plane that passes through the point with position vector   A  =  5  i  +  j  +  3  k and the line       l    1    =  2  i  −  8  j  −  17  k  +  λ  (  i  −  3  j  −  4  k  )I found the normal by finding the cross product of   A  ∗  [  (  5  i  +  j  +  3  k  )  −  (  2  i  −  8  j  −  17  k  )  ] which turned out to be       n    1    =  −  24  i  −  32  j  +  18  k. Then I used   r  ⋅  n  =  a  ⋅  n  →  (  5  i  +  j  +  3  k  )  ⋅  (  −  24  i  −  32  j  +  18  k  )  =  −  98However, it turns out that the above value should be 49 rather than −98. I realize that it can be divided by -2 to get the answer; however, I don&#039;t understand why this happens.

farbhas3t · Accepted Answer

−98  is correct if you use       n    1    =  −  24  i  −  32  j  +  18  k as normal vector. Then the equation of the plane is                    (1)                    −        24        x        −        32        y        +        18        z        =        −        98        .            But you can divide the normal vector by −2, obtaining       n    2    =  12  i  +  16  j  −  9  k. For such normal vector the correct value is 49 and the equation of the plane is                    (2)                    12        x        +        16        y        −        9        z        =        49        .            Of course the equations (1) and (2) are equivalent and they represents the same plane.

I need to find the equation of a plane that passes through the point with position vector A=5i+j+3k and the line l_1=2i−8j−17k+lambda(i−3j−4k).

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