Equations of lines and planes
1. Find a vector and parametric equations for the line that passes through (4,2) and is parallel to v = <-1,5>. Then find 2 other points on that line.
2. Find parametric equations for the line segment joining points P(2, -4, -1) and Q(5, 0, 7). Where does this line intersect the xy-plane?
To solve these two questions, we need to apply concepts from vectors and parametric equations, which describe lines in space.
1. Finding the vector and parametric equations for a line through (4,2) parallel to v = <-1,5>:
A line passing through a point can be represented by using a direction vector. In this case, the direction vector is given as v = <-1,5>, which tells us the direction the line travels in. Since the line passes through the point (4,2), we can use that as the starting point.
To find the parametric equations, we express the x and y coordinates of points on the line in terms of a parameter, often called t, which moves along the line. The general formula for parametric equations in 2D involves adding the direction vector multiplied by t to the point the line passes through. In this case:
The x-coordinate of points on the line starts at 4 and moves along by -1 each step.
The y-coordinate starts at 2 and moves along by 5 each step.
The parametric equations describe all points on the line. To find two other points on the line, plug in different values of t, for example, t = 1 and t = -1, and calculate the corresponding coordinates.
2. Parametric equations for the line segment joining points P(2, -4, -1) and Q(5, 0, 7):
In this case, we want to describe the line segment between two points P and Q. To do this, we first find the direction vector between the two points by subtracting P from Q. This vector will describe the direction of the line segment.
Once we have the direction vector, the parametric equations for the line segment can be written by starting at P and adding multiples of the direction vector. The parameter t will range from 0 to 1, where t = 0 corresponds to point P and t = 1 corresponds to point Q.
To find where the line intersects the xy-plane, we need to find when the z-coordinate of the line equals 0. Set the z-equation from the parametric equations to 0 and solve for t. Once you have t, substitute that value into the x and y parametric equations to find the coordinates of the intersection point in the xy-plane.