Answer: Are Pln & Line Othogonal/Parallel? Solution to (-x+2z=10)

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The discussion focuses on determining the relationship between the plane defined by the equation (-x + 2z = 10) and the line represented by the vector equation r = <5, 2 - t, 10 + 4t>. The normal vector of the plane is identified as <-1, 0, 2>, while the direction vector of the line is <0, -1, 4>. To establish orthogonality, the dot product of these vectors must equal zero, while parallelism requires that one vector is a scalar multiple of the other. The analysis concludes that the line and plane are neither orthogonal nor parallel.

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Homework Statement



Determin if the plan given by (-x+2z=10) and the line given by r=<5,2-t,10+4t> are othogonal,parllel or neither??

What is the solution?
 
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I'm not going to give you the solution- the point is for you to find the solution. I will suggest this: your line is given by the vector equation r= <5, 2, 10>+ t<0,-1,4> so <0, -1, 4> is a vector pointing in the direction of that line. Also, the plane -x+ 0y+ 2z= 10 has <-1, 0, 2> as normal vector (that is <-1, 0, 2> is perpendicular to the plane).

What must be true of those two vectors in order that the line and plane be perpendicular? parallel? Are those true?
 

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