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Section 1-3 : Equations of Planes

7. Determine if the line given by \(\vec r\left( t \right) = \left\langle {4 + t, - 1 + 8t,3 + 2t} \right\rangle \) intersects the plane given by \(2x - y + 3z = 15\) or show that they do not intersect.

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Start Solution

If the line and the plane do intersect then there must be a value of \(t\) such that if we plug that \(t\) into the equation of the line we’d get a point that lies on the plane. We also know that if a point \(\left( {x,y,z} \right)\) is on the plane the then the coordinates will satisfy the equation of the plane.

Show Step 2

If you think about it the coordinates of all the points on the line can be written as,

\[\left( {4 + t, - 1 + 8t,3 + 2t} \right)\]

for all values of \(t\).

Show Step 3

So, let’s plug the “coordinates” of the points on the line into the equation of the plane to get,

\[2\left( {4 + t} \right) - \left( { - 1 + 8t} \right) + 3\left( {3 + 2t} \right) = 15\] Show Step 4

Let’s solve this for \(t\) as follows,

\[18 = 15\,\,??\] Show Step 5

Hmmm…

So, either we’ve just managed to prove that 18 and 15 are in fact the same number or there is something else going on here.

Clearly 18 and 15 are not the same number and so something else must be going on. In fact, all this means is that there is no \(t\) that will satisfy the equation we wrote down in Step 3. This in turn means that the line and plane do not intersect.