Chapter 2

1. Chapter 2 Section 2.4 Lines and Planes in Space

2. z Equations of Lines To form the equation of a line in 3 dimensions we need two pieces of information. A point on the line and a vector d called the direction vector that tells us the orientation (“slope“) of the line. Once you are on the line to get to another point it is just a scalar multiple of d (i.e. ). The location of another point y x Parametric Form Vector Form Example Find the vector equation passing through the point and parallel to the line with equation given to the right. We already know the point is on the line so we need to find the direction d. Since the lines are parallel they will have parallel direction vectors. The coefficients of t form the direction vector d.

3. Example Find the vector equation of the line passing through the points and . A direction vector d for the line can be found by finding the vector from the first point to the second. Example Find the equation of the line that is perpendicular to both of the vectors and passing through the point . To get a direction vector d that is perpendicular to both vectors u and v we can use the cross product. To get the equation for the line we plug the vector d and the point into the equation.

4. Example Determine if the lines and given to the right intersect. If they intersect find the point of intersection. The tricky part here is to realize that the value for t that gives the point of intersection in might not be the same value of t that give the point of intersection in . Choose a different independent variable for , say u. This gives a system of equations with the variables t and u. Solve this system for t and u. This can be done in many ways here we equate the x and y components. Substituting the values for t and u into their equations we see the lines intersect at the point .

5. Equations of Planes A normal vector N for a plane is a vector that is orthogonal to every vector in the plane. This tells how the plane is oriented, it is like the “slope“ of the plane. To get the equation we form the vector from the known point on the plane to the variable point and dot it with N. The result must be zero. z y x Plane with normal and point is: Example A plane is perpendicular to the line given to the right and intersects the line at a point with x-coordinate 4. Find the equation of the plane. To find a point on the line set the x-coordinate equal to 4 and solve for t. Plug that value into l to find the point. The direction vector l can be the normal vector N for the plane. Equation:

6. Example Remember that 3 points determine a plane. Find the equation of the plane that contains the points , and . We can use any of the points P,Q, or R as a point on the plane. To find a normal vector N form the vectors in the plane and . A normal vector will be perpendicular to both of them, so take the cross product and let . We use the point to be the point and substitute in formula. Equation of plane: Check at each point:

7. Example Find the equation of the line of intersection between the two planes whose equations are given to the right. One approach is to find two points on the line and use them to determine the direction vector d for the line. In order to do this we need to solve a system of equations and get 2 particular solutions. For this we let and in the general solution. This gives the following points: The direction vector Equation: General Solution Check: