Line Through a Point Perpendicular to a Plane:

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Find the symmetric and parametric equations of the line through point A(4,7,5) perpendicular to the plane .

Let's graph this in steps. First, we'll find the intercepts of the plane on the three axes.
How do we find the x-intercept, ?

Since this intercept is on the x-axis, it must have .

Use these values in the equation of the plane to find

We get

Plot this on paper, using the usual 3-D axes with the x-axis out of the plane of the diagram. Then check your work by clicking "Next".

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Find in a similar way.

Plot this point, and then click "Next" to check.

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Find in a similar way.

Plot this point. Then click "Next" to check.

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Connect these points to show a portion of the plane. Then click "Next" to check your graph.

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Add point A to the diagram. Then click "Next" to check.

Note the use of color for the coordinates of the point. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

What do we need to do next to find the desired line?

We need a vector perpendicular to the plane.

How can we find such a vector?

We can use the plane's normal vector, n.

How do we get its components?

The coefficients of in the equation of the plane are the components of the normal vector.

Use this information to express the normal vector.

How do we use n.

We can get the parametric equations of the line by using the components of n and the coordinates of A(4,7,5).

Use this information to express the parametric equation for the x-coordinate of any point on the line.

Express the y-coordinate in a similar way.

Express the z-coordinate in a similar way.

Now we shall go on to find the symmetric equations for the line. How do we do that?

We solve each of the parametric equations for t.

Do that for x.

Do the same for y.

Do the same for z.

Let's find the coordinates of the point, B, where our line intersects the plane.
How are the coordinate of that point
related to the equation of the plane
and the parametric equations of the line?

The coordinates must satisfy all of the equations.

Set up those equations in terms of the coordinates.

Equation 1
Equation 2

Is the same for each equation?

Yes.

How do we solve?

We substitute the expressions from Equation 2 into Equation 1. Then we will have an equation involving only , which we can solve for.

Do that.

We get

Do the multiplication.

This becomes

Collect like terms.

Solve this.

We get

Use this value of t in the parametric equations to find the coordinates of the point of intersection.

What can we conclude?

The point of intersection, B, of the line with the plane has coordinates (0,1,2).

Add this point to your paper plot. Then click "Next" to check.

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As a further check, use t = +1 in the parametric equations to determine another point, C.

Add this point to your paper plot. Then click "Next" to check.

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The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

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