Topic 1.6: Lines and Planes

Transcription

1 Math 275 Notes (Ultman) Topic 1.6: Lines and Planes Textbook Section: 12.5 From the Toolbox (what you need from previous classes): Plotting points, sketching vectors. Be able to find the component form a vector given two points. Vector operations: vector addition, scalar multiplication, dot product, and cross product. Know how the dot and cross products are related to orthogonality of vectors. Learning Objectives (New Skills) & Important Concepts Learning Objectives (New Skills): Find a vector equation r = r 0 + tv and parametric equations for a line, given a direction vector v and a point P 0 the line passes through. Find a direction vector v for a line given a vector equation or parametric equations for the line, or given a graph of the line. Use the direction vectors of two lines to determine whether or not the lines are parallel.

2 Planes: Find the vector equation n P 0 P = 0 of a plane, given a normal vector n and a point P 0 the plane passes through. Use the vector equation to find the algebraic equation ax + by + cz = d. Find an equation of a plane given three points in the plane. Find a normal vector n to a plane given the algebraic equation ax + by + cz = d of the plane. Use the normal vectors of two planes to determine whether or not the planes are parallel. Important Concepts: To find the vector equation of a line, r = r 0 + tv, you need two pieces of information: a vector v giving the direction of the line, and the coordinates of a point P 0 on the line. The vector r 0 is the position vector of the point P 0 (that is, r 0 = OP ). Parallel lines have parallel direction vectors. Planes: To find the vector equation of a plane, n P 0 P = 0, you need two pieces of information: a vector n normal to the plane, and the coordinates of a point P 0 on the plane. If ax + by + cz = d is an algebraic equation of a plane, then n = a, b, c is a normal vector to a plane. If n is normal to a plane, then all scalar multiples cn are also normal to the plane. Parallel planes have the same normal vectors. The equation of a plane can be found beginning with three noncollinear points on the plane, using the cross product to produce a normal vector. 2

3 The Big Picture In previous math classes, the main way of representing lines in R 2 has been the equation y = mx + b. We will now learn a way of representing lines using a vector equation: r = r 0 + tv This vector equation works in any dimension R 2, R 3, R 4, etc. The vector v gives the direction of the line. The vector r 0 is the position vector of a point P 0 on the line; in other words, r 0 = OP is the vector from the origin O to the point P 0. Planes: A plane ax + by + cy = d in R 3 equation: can also be represented using a vector n P 0 P = 0 Where n = a, b, c is normal to the plane, P 0 = (x 0, y 0, z 0 ) is a point on the plane, and P 0 P = x x 0, y y 0, z z 0 is the vector from the fixed point P0 to the arbitrary point P = (x, y, z). Lines and Planes: Note that the vector equations of both lines and planes require a point and a vector. The point is a point on the line or plane. The vector provides the orientation of the line or plane that is, how it is positioned in space. For a line, the vector is parallel to the line. For a plane, the vector is normal to the plane. Parallel lines have parallel directional vectors. Parallel planes have parallel normal vectors. 3

4 More Details Two reasons why using a vector equation to represent lines is useful: The slope-intercept equation of a line y = mx +b only works for lines in the xy-plane (R 2 ). It does not work for lines in higher dimensions, like 3-space (R 3 ). A line can be represented by infinitely many vector equations. Why is this useful? Consider the following. The vector equation r = r 0 + tv can be used to represent the path of an object that travels along a line with velocity v, and passes through the terminal point of the vector r 0 at time t = 0. Clearly, different objects can travel along the same path with different velocities, and may pass through different points at t = 0. There are similarities between the slope-intercept equation of a line y = mx + b and the vector equation r = r 0 + tv: Both require two pieces of information. For the slope-intercept equation, you need the slope m and the y-intercept b. For the vector equation, you need a vector v for the direction of the line, and a point P 0 on the line. Both require information about the direction of the line. In the slope-intercept equation, the direction is given by the slope m. In the vector equation, the direction is given by a vector v. Both require a point P 0 on the line. In the slope-intercept equation, the point P 0 = (0, b) gives the y-intercept, y = b. In the vector equation, the fixed vector r 0 is the position vector of the point P 0 on the line. To parameterize a line means to find a vector equation r = r 0 + tv for the line. The parameter is the variable in this case, t. (An application of parameterizations: Finding the location of an object at time t.) 4

5 The coordinate functions or parametric equations of a line described by a vector equation are the functions that give the components of the vector equation. If the parameter is t, these components will all be functions of the variable t. If the vector equation represents the location at time t of an object traveling along a line, then the parametric equations give the x, y, and z coordinates of the object at time t. Planes: A vector is normal to a plane if it is orthogonal to every vector in the plane. This leads to the vector equation of a plane. If P 0 = (x 0, y 0, z 0 ) is a fixed point in the plane, then the point P = (x, y, z) will also be in the plane if n P 0 P = 0. There are infinitely many vectors that are normal to a plane. If n 0 is normal to a plane P, then every scalar multiple cn is also normal to the plane. The line parameterized by r = tn is called a normal line of P. A normal line of a plane is perpendicular to the plane. Two planes in R 3 are parallel if and only if they have the same normal line. So a vector that is normal to one plane will also be normal to any other plane that is parallel to it. There is another way of writing the vector equation of the plane. If r 0 = x 0, y 0, z 0 is the position vector of the point P0, and r = x, y, z is the general position vector, then P 0 P = x x 0, y y 0, z z 0 = r r 0, so: n P 0 P = 0 n (r r 0 ) = 0 n r n r 0 = 0 n r = n r 0 5

6 To see the relationship between a vector equation of a plane and an algebraic equation of the plane, let n = a, b, c for some constants a, b, and c (not all equal to zero), and let P 0 = (x 0, y 0, z 0 ) be a fixed point on a plane (so x 0, y 0, and z 0 are constants). Then: n P 0 P = 0 n r = n r 0 a, b, c x, y, z = a, b, c x0, y 0, z 0 ax + by + cz = d where: d = a, b, c x 0, y 0, z 0 = ax0 + by 0 + cz 0 From the above, you can see that starting with an algebraic equation ax + by + cz = d, a normal vector to the plane is n = a, b, c. In other words, the x, y, and z coefficients in the algebraic equation of a plane give the î, ĵ, and ˆk components of a vector normal to the plane. Suppose u and v determine a plane. Since u v is orthogonal to both u and v, then u v is normal to the plane spanned by u and v. This can be used to find the equation of a plane given three non-collinear points on the plane. 6