topics_13_5_13_6_1 - 13.5 Equations of lines and planes...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
13.5. Equations of lines and planes Vector equation of a line. Consider a line L with a point P 0 ( x 0 , y 0 , z 0 ) from the line and a vector v parallel to the line. Let P ( x, y, z ) be an arbitrary point from the line L . Denote by r 0 the position vector corre- sponding to P 0 and by r the position vector corresponding to P . Then, there is a scalar t such that r = r 0 + t v , -∞ < t < , where t is a parameter and equivalently in terms of components of vectors: h x , y , z i = h x 0 , y 0 , z 0 i + t h v 1 , v 2 , v 3 i . The above equation is called a vector equation of the line L . v is called a directional vector of L and its components v = h v 1 , v 2 , v 3 i are called direction numbers of L . Parametric equation of a line. If we equal the corresponding com- ponents in the above vector equation we obtain parametric equations of a line: x = x 0 + v 1 t y = y 0 + v 2 t z = z 0 + v 3 t. This is one-parameter ( t is the parameter) representation of the line L passing through the points P 0 ( x 0 , y 0 , z 0 ) and parallel to the directional vector v . The vector equation and the parametric equations of a line are not unique. They depend on the choice of P 0 and v. Exercise. Suppose that the line L is on the xy -plane. Then what will be the form of the above parametric equations? Symmetric equation of a line. Eliminating the parameter t we obtain symmetric equations of the line L : x - x 0 v 1 = y - y 0 v 2 = z - z 0 v 3 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 6/page 838. Find a vector equation, parametric equations, and symmetric equations for the line passing through the origin and the point (1 , 2 , 3). Problem 10/page 838. Find parametric equations and symmetric equations for the line passing through the point (2 , 1 , 0) and perpen- dicular to both i + j and j + k . Problem 12/page 838. Find parametric equations and symmetric equations for the line of intersection of the planes x + y + z = 1 and x + z = 0. Problem 16/page 838. (a) Find parametric equations, and sym- metric equations for the line passing through the point (2 , 4 , 6) that is perpendicular to the plane x - y + 3 z = 7. (b) In what points does the line intersects the coordinate planes? Problems 19,20/page 838. Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. Find the point of intersection if they intersect.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern