Calculus III: Find a line perdendicular to XY-plane?

In summary, the problem is to find an equation for a line passing through the point P = (1, 0, −3) and perpendicular to the xy-plane. This can be achieved by using the vector equation r = r0 + tv, where r0 is the position vector <1,0,-3> and v is a direction vector perpendicular to the xy-plane, which in this case is <0,0,1>. This can also be written as r=<1,0,t-3>. Other forms such as r x a = b, for constant vectors a and b, could also be used.
  • #1
whig4life
14
0
Homework Statement [/b]
"Find an equation for the line through the point P = (1, 0, −3) and perpendicular
to the xy-plane,"

obviously this includes vector <0, 0, 1>

I am in Calc III and need help understanding how to do this TYPE of problem. Please include step-by-step instructions and any concepts used. Thank you.
 
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  • #2
Why don't you try drawing a sketch? If a line is perpendicular to the x-y plane, how do the values of x and y change when the z-coordinate varies?
 
  • #3
whig4life said:
"Find an equation for the line through the point P = (1, 0, −3) and perpendicular
to the xy-plane,"
obviously this includes vector <0, 0, 1>
It does? Did you mean <1, 0, 0>?
More generally, what can you say about the values of x and y on that line?
I assume it's a vector equation you're after?
 
  • #4
haruspex said:
More generally, what can you say about the values of x and y on that line?
I assume it's a vector equation you're after?

Yes it's a vector equation I am after.

my thought is: r = r0 + tv meaning r=<1,0,-3> + t<0,0,1>

I just don't know if I'm right. Need clarification
 
  • #5
whig4life said:
Yes it's a vector equation I am after.

my thought is: r = r0 + tv meaning r=<1,0,-3> + t<0,0,1>
Sure, or just r=<1,0,t-3>. But that's a parametric equation, which might not be what's wanted. Another form might be r x a = b, for some constant vectors a and b.
 
  • #6
haruspex said:
Sure, or just r=<1,0,t-3>. But that's a parametric equation, which might not be what's wanted. Another form might be r x a = b, for some constant vectors a and b.

Still highly confused.
 
  • #7
Can you find constant vectors a and b such that the equation r x a = b implies r is of the form <1,0,*>?
 
  • #8
whig4life said:
Yes it's a vector equation I am after.

my thought is: r = r0 + tv meaning r=<1,0,-3> + t<0,0,1>

I just don't know if I'm right. Need clarification

Yes, that's exactly right. <1,0,-3> is a position vector to the point and <0,0,1> is a correct direction vector.
 
Last edited:

1. What is the XY-plane in Calculus III?

The XY-plane, also known as the Cartesian plane, is a two-dimensional coordinate system used in Calculus III to graph and analyze functions. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The point of intersection of the two axes is called the origin, denoted by (0,0).

2. How do you find a line perpendicular to the XY-plane?

A line perpendicular to the XY-plane is a line that is parallel to the z-axis. To find this line, we need to consider the equation of a plane in three-dimensional space, which is given by Ax + By + Cz + D = 0. Since we want the line to be parallel to the z-axis, we set A and B equal to 0. This gives us the equation Cz + D = 0, which represents any line perpendicular to the XY-plane.

3. What is the equation of a line perpendicular to the XY-plane passing through a given point?

To find the equation of a line perpendicular to the XY-plane passing through a given point (x0,y0,z0), we first need to find the slope of the line. This slope is given by the ratio of the change in z-coordinates to the change in x-coordinates, which is equal to -1. Using this slope and the given point, we can write the equation of the line as z - z0 = -1(x - x0). Simplifying this equation gives us the final equation of the line perpendicular to the XY-plane passing through the given point as z = -x + (x0 + z0).

4. How can we visualize a line perpendicular to the XY-plane?

To visualize a line perpendicular to the XY-plane, we can imagine it as a line that extends infinitely in the z-direction while remaining parallel to the x and y-axes. Alternatively, we can also use a 3D graphing software or plot the line on a 3D coordinate plane to get a better understanding of its orientation and its relationship with the XY-plane.

5. What are the applications of finding a line perpendicular to the XY-plane?

Finding a line perpendicular to the XY-plane is a fundamental concept in Calculus III and has various applications in real-life scenarios. For example, in engineering and architecture, this concept is used to determine the orientation of structures in three-dimensional space. It is also used in physics to understand the motion of objects in three dimensions and in computer graphics to create 3D models and animations.

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