# Solutions to Problem Set 5

Math 311-01/02

2-1-2018

[Lines and planes problems]

1. Find the parametric and the symmetric equations for the line which passes through the point and is parallel to the vector .

The parametric equations are

Solve each of these equations for t and equate the results to get the symmetric equations:

2. Find parametric equations for the line which passes through the point and is parallel to the line

The given line is parallel to the vector , so the same is true for the line I'm constructing. The line is

3. Find the parametric equation of the line which passes through the points and .

The line is parallel to . The equation of the line can be written using either P or Q:

4. Find the parametric equation of the line which passes through the point and is perpendicular to both of the vectors and .

The line is parallel to . Its equation is

5. Find the equation of the plane which contains the point and is perpendicular to the vector .

6. Find the equation of the plane containing the points , , and .

The vectors and lie in the plane, so their cross product is perpendicular to the plane:

Using P as the point in the plane, the plane is

7. Sketch the following plane by finding its x, y, and z-intercepts:

Setting gives , or .

Setting gives , or .

Setting gives .

8. Find the equation of the plane which contains the point and is parallel to the plane

The given plane is perpendicular to the vector . Since the planes are parallel, the plane I want is also perpendicular to the vector . the plane is

The struggle to understand is our only advantage over this madness. - Ta-Nehisi Coates

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