Figure 12.5.3. Distance from a point to a plane. Show
Figure 12.5.4. Distance from a point to a line. Exercises 12.5You can use Sage to compute distances to lines and planes, since this just involves vector arithmetic that we have already seen. Of course, you can also use Sage to do some of the computations involved in finding equations of planes and lines. Ex 12.5.1 Find an equation of the plane containing $(6,2,1)$ and perpendicular to $\langle 1,1,1\rangle$. (answer) Ex 12.5.2 Find an equation of the plane containing $(-1,2,-3)$ and perpendicular to $\langle 4,5,-1\rangle$. (answer) Ex 12.5.3 Find an equation of the plane containing $(1,2,-3)$, $(0,1,-2)$ and $(1,2,-2)$. (answer) Ex 12.5.4 Find an equation of the plane containing $(1,0,0)$, $(4,2,0)$ and $(3,2,1)$. (answer) Ex 12.5.5 Find an equation of the plane containing $(1,0,0)$ and the line $\langle 1,0,2\rangle + t\langle 3,2,1\rangle$. (answer) Ex 12.5.6 Find an equation of the plane containing the line of intersection of $x+y+z=1$ and $x-y+2z=2$, and perpendicular to the plane $2x+3y-z=4$. (answer) Ex 12.5.7 Find an equation of the plane containing the line of intersection of $x+2y-z=3$ and $3x-y+4z=7$, and perpendicular to the plane $6x-y+3z=16$. (answer) Ex 12.5.8 Find an equation of the plane containing the line of intersection of $x+3y-z=6$ and $2x+2y-3z=8$, and perpendicular to the plane $3x+y-z=11$. (answer) Ex 12.5.9 Find an equation of the line through $(1,0,3)$ and $(1,2,4)$. (answer) Ex 12.5.10 Find an equation of the line through $(1,0,3)$ and perpendicular to the plane $x+2y-z=1$. (answer) Ex 12.5.11 Find an equation of the line through the origin and perpendicular to the plane $x+y-z=2$. (answer) Ex 12.5.12 Find $a$ and $c$ so that $(a,1,c)$ is on the line through $(0,2,3)$ and $(2,7,5)$. (answer) Ex 12.5.13 Explain how to discover the solution in example 12.5.5. Ex 12.5.14 Determine whether the lines $\langle 1,3,-1\rangle+t\langle 1,1,0\rangle$ and $\langle 0,0,0\rangle+t\langle 1,4,5\rangle$ are parallel, intersect, or neither. (answer) Ex 12.5.15 Determine whether the lines $\langle 1,0,2\rangle+t\langle -1,-1,2\rangle$ and $\langle 4,4,2\rangle+t\langle 2,2,-4\rangle$ are parallel, intersect, or neither. (answer) Ex 12.5.16 Determine whether the lines $\langle 1,2,-1\rangle+t\langle 1,2,3\rangle$ and $\langle 1,0,1\rangle+t\langle 2/3,2,4/3\rangle$ are parallel, intersect, or neither. (answer) Ex 12.5.17 Determine whether the lines $\langle 1,1,2\rangle+t\langle 1,2,-3\rangle$ and $\langle 2,3,-1\rangle+t\langle 2,4,-6\rangle$ are parallel, intersect, or neither. (answer) Ex 12.5.18 Find a unit normal vector to each of the coordinate planes. Ex 12.5.19 Show that $\langle 2,1,3 \rangle + t \langle 1,1,2 \rangle$ and $\langle 3, 2, 5 \rangle + s \langle 2, 2, 4 \rangle$ are the same line. Ex 12.5.20 Give a prose description for each of the following processes: a. Given two distinct points, find the line that goes through them. b. Given three points (not all on the same line), find the plane that goes through them. Why do we need the caveat that not all points be on the same line? c. Given a line and a point not on the line, find the plane that contains them both. d. Given a plane and a point not on the plane, find the line that is perpendicular to the plane through the given point. Ex 12.5.21 Find the distance from $(2,2,2)$ to $x+y+z=-1$. (answer) Ex 12.5.22 Find the distance from $(2,-1,-1)$ to $2x-3y+z=2$. (answer) Ex 12.5.23 Find the distance from $(2,-1,1)$ to $\langle 2,2,0\rangle+t\langle 1,2,3\rangle$. (answer) Ex 12.5.24 Find the distance from $(1,0,1)$ to $\langle 3,2,1\rangle+t\langle 2,-1,-2\rangle$. (answer) Ex 12.5.25 Find the distance between the lines $\langle 5,3,1\rangle+t\langle 2,4,3\rangle$ and $\langle 6,1,0\rangle+t\langle 3,5,7\rangle$. (answer) Ex 12.5.26 Find the distance between the lines $\langle 2,1,3\rangle+t\langle -1,2,-3\rangle$ and $\langle 1,-3,4\rangle+t\langle 4,-4,1\rangle$. (answer) Ex 12.5.27 Find the distance between the lines $\langle 1,2,3\rangle+t\langle 2,-1,3\rangle$ and $\langle 4,5,6\rangle+t\langle -4,2,-6\rangle$. (answer) Ex 12.5.28 Find the distance between the lines $\langle 3,2,1\rangle+t\langle 1,4,-1\rangle$ and $\langle 3,1,3\rangle+t\langle 2,8,-2\rangle$. (answer) Ex 12.5.29 Find the cosine of the angle between the planes $x+y+z=2$ and $x+2y+3z=8$. (answer) Ex 12.5.30 Find the cosine of the angle between the planes $x-y+2z=2$ and $3x-2y+z=5$. (answer)
 First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow. ../geometry/images/geom-line-equn.js We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is: Example: The point (12,5) is 12 units along, and 5 units up StepsThere are 3 steps to find the Equation of the Straight Line :
Step 1: Find the Slope (or Gradient) from 2 PointsWhat is the slope (or gradient) of this line? We know two points:
The slope is the change in height divided by the change in horizontal distance. Looking at this diagram ... Slope m = change in ychange in x = yA − yBxA − xB In other words, we:
Like this: m = change in y change in x = 4−3 6−2 = 1 4 = 0.25 It doesn't matter which point comes first, it still works out the same. Try swapping the points: m = change in y change in x = 3−4 2−6 = −1 −4 = 0.25 Same answer. Step 2: The "Point-Slope Formula"Now put that slope and one point into the "Point-Slope Formula" Start with the "point-slope" formula (x1 and y1 are the coordinates of a point on the line): y − y1 = m(x − x1) We can choose any point on the line for x1 and y1, so let's just use point (2,3): y − 3 = m(x − 2) We already calculated the slope "m": m = change in ychange in x = 4−36−2 = 14 And we have: That is an answer, but we can simplify it further. Step 3: Simplify
Start with:y − 3 = 14(x − 2) Multiply 14 by (x−2):y − 3 = x4 − 24 Add 3 to both sides:y = x4 − 24 + 3 Simplify:y = x4 + 52 And we get: Which is now in the Slope-Intercept (y = mx + b) form.
Let us confirm by testing with the second point (6,4): y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4 Yes, when x=6 then y=4, so it works! Another ExampleThe Big ExceptionThe previous method works nicely except for one particular case: a vertical line:
7270, 525, 526, 1165, 1166, 7291, 7292, 7300, 7301, 7302 Copyright © 2022 Rod Pierce |
What is the equation of the line which passes through the point 2 3 and (- 4 1 is?
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