The parametric equation of a line, x = x0 + at, y = y0 + bt and z = z0 + ct The point of intersection is a common point of a line and a plane. This is Mathepower. If we take the parameter at being one of the coordinates, this usually simplifies the algebra. 0. We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication – we have to introduce a parameter. Can you please explain what is the issue? Do you mean lines or line segments? Vote. What if we keep the same line, but modify the plane equation to be $$x + 2y - 2z = -1$$? If they do intersect, determine whether the line is contained in the plane or intersects it in a single point. We can verify this by putting the coordinates of this point into the plane equation and checking to see that it is satisfied. Here you can calculate the intersection of a line and a plane (if it exists). Can i see some examples? So the point of intersection can be determined by plugging this value in for $$t$$ in the parametric equations of the line. This means that this line does not intersect with this plane and there will be no point of intersection. Have questions or comments? There are three possibilities: The line could intersect the plane in a point. Or the line could completely lie inside the plane. You say "lines" but you say they have length. 0 ⋮ Vote. Even if this plane and line is not intersecting, it shows check=1 and intersection point I =[-21.2205 31.6268 6.3689]. Or the line could completely lie inside the plane. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Legal. These lines are parallel when and only when their directions are collinear, namely when the two vectors and are linearly related as u = av for some real number a. Follow 40 views (last 30 days) Stephanie Ciobanu on 9 Nov 2017. If the line does intersect with the plane, it's possible that the line is completely contained in the plane as well. Notice that we can substitute the expressions of $$t$$ given in the parametric equations of the line into the plane equation for $$x$$, $$y$$, and $$z$$. Distinguishing these cases and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection. Collecting like terms on the left side causes the variable $$t$$ to cancel out and leaves us with a contradiction: Since this is not true, we know that there is no value of $$t$$ that makes this equation true, and thus there is no value of $$t$$ that will give us a point on the line that is also on the plane. Can i see some examples? Missed the LibreFest? The line case is a lot easier because any two non-parallel lines in an x,y plane will intersect somewhere, not so with segments – user316117 Dec 28 '15 at 18:31 Finally the code uses the adjusted values of t1 and t2 to find those closest points. Finally, if the line intersects the plane in a single point, determine this point of intersection. Now that we have examined what happens when there is a single point of intersection between a line and a point, let's consider how we know if the line either does not intersect the plane at all or if it lies on the plane (i.e., every point on the line is also on the plane). We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication – we have to introduce a parameter. Simply enter your exercise and it will be solved step by step. Line: x = 2 − t Plane: 3 x − 2 y + z = 10 y = 1 + t z = 3 t. Watch the recordings here on Youtube! So the point of intersection of this line with this plane is $$\left(5, -2, -9\right)$$. 0 Comments . No. If they do intersect, determine whether the line is contained in the plane or intersects it in a single point. As shown in the diagram above, two planes intersect in a line. As shown in the diagram above, two planes intersect in a line. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Example $$\PageIndex{9}$$: Other relationships between a line and a plane, \begin{align*} \text{Line:}\quad x &=1 + 2t & \text{Plane:} \quad x + 2y - 2z = 5 \\[5pt] y &= -2 + 3t \\[5pt] z &= -1 + 4t \end{align*}\nonumber. Finally, if the line intersects the plane in a single point, determine this point of intersection. In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. The code then adjusts t1 and t2 so they are between 0 and 1. Sign in to comment. A given line and a given plane may or may not intersect. Do a line and a plane always intersect? As shown in the diagram above, two planes intersect in a line. We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication – we have to introduce a parameter. Example $$\PageIndex{8}$$: Finding the intersection of a Line and a plane. Check: $$3(5) - 2(-2) + (-9) = 15 + 4 - 9 = 10\quad\checkmark$$. This is equivalent to the conditions that all . Here: $$x = 2 - (-3) = 5,\quad y = 1 + (-3) = -2, \,\text{and}\quad z = 3(-3) = -9$$. Therefore, coordinates of intersection must satisfy both equations, of the line and the plane. Commented: Star Strider on 9 Nov 2017 Accepted Answer: Star Strider. 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