Lines and Planes in Space Lines in Space

Lines and Planes in Space

Lines in Space

Recall that, In a plane, slope is used to determine the equation of the line. In space, it’s more convenient to use vectors to find equation of a line.

z L y x

Vector Equation of a line

Equation: r = a + tb

Example 1 a. Find a vector equation and parametric equations for the line that passes through the point (5, 1, 3) and is parallel to the vector i + 4 j – 2 k. b. Find two other points on the line.

Do lines have a unique vector (parametric) equation?

Remark • The vector equation and parametric equations of a line are not unique. – If we change the point or the parameter or choose a different parallel vector, then the equations change.

Example 2 a. Find parametric equations and symmetric equations of the line that passes through the points A(2, 4, – 3) and B(3, – 1, 1). b. At what point does this line intersect the xy-plane?

Parallel Perpendicular skew lines

When do we call two lines are 1. Parallel? 2. perpendicular? 3. Skew?

Example-3 Show that the following lines are parallel.

Example 4 Show that the lines L 1 and L 2 with parametric equations x = 1 + t y = – 2 + 3 t z = 4 – t x = 2 s y=3+s z = – 3 + 4 s are skew lines. – That is, they do not intersect and are not parallel, and therefore do not lie in the same plane.

• How do we draw skew lines? L 1 and L 2 are skew lines.

Today’s potential Question Can we say two lines, on different planes, are perpendicular if the dot product of their direction vector is zero? My answer is: NO Let’s argue about it tomorrow in class! https: //www. brightstorm. com/math/precalc ulus/vectors-and-parametricequations/perpendicular-parallel-and-skewlines-in-space/

Class work • Homework………. Finish off Class work!