1
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1answer
21 views

Why 2 equations of the form F(x,y,z) = 0 for one 3D curve

It says in my analysis 2 book that a curve is given by $F_1(x,y,z) = 0$ and $F_2(x,y,z) = 0$. Why do we need two equations of $x,y,z$ To define a curve in 3D, shouldn't one be enough?
2
votes
2answers
49 views

Two and Three Variable Limit Questions

Find the following limits, if they exist. $$\lim_{x,y\rightarrow 0,0}\frac{x^2 + \sin^2 y}{\sqrt{x^2+y^2}}$$ I believe we're suppose to use the squeeze theorem on this first one above. Possibly ...
1
vote
1answer
184 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
0
votes
1answer
99 views

Volume between paraboloid and plane

I need to find the volume of the finite region enclosed between the surface $$ y = 1 - x^2 - 4z^2 $$ and the plane $$y = 0$$ Here's what I've done: $$ \int\int ...
1
vote
1answer
46 views

Volume of the Region bounded by $y = 2x^2 +2z^2$ and the plane $y=8$

I have the find the volume of the region bounded by the paraboloid $y = 2x^2 +2z^2$ and the plane $y=8$. Is the volume (using triple integrals) just ...
0
votes
1answer
194 views

How to find the curve of intersection of a ellipsoid and a plane?

Let $C$ be the curve of intersection of the ellipsoid $x^2+2y^2+3z^2=39$ and the plane $3x+y-7z=0$. Find the parametric equations for the tangent line to $C$ at $(5,-1,2)$. I don't know how to find ...
1
vote
2answers
113 views

Finding the equation and plotting a plane using 3 points

restart; with(plots): with(VectorCalculus): I have 3 points in a plane defined in Maple as: ...
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votes
1answer
264 views

Equation of plane through intersection of planes and parallel to line

Find the equation of the plane through the intersection of the planes of $x-2y+z=1$ and $2x+y+z=8$ and parallel to the line: $\frac{x-3}{1} = \frac{y-1}{2} = \frac{z-2}{1} $ I'm facing difficulties ...
0
votes
1answer
32 views

Equation of a plane containing a point and a line

Find the equation of the plane containing the point (0, 7, -7) and the line $\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$ I'm not sure how to tackle this question, since the equation of the line ...
2
votes
1answer
40 views

Question about reexpressing the dot product

Suppose that I have two arbitrary 3-dimensional vectors, $\vec{a}$ and $\vec{b}$. By the definition of the dot product, I can write $$\vec{a} \cdot \vec{b} = \left|\vec{a}\right| ...
1
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0answers
76 views

3d Implicit Trigonometry help?

I'm trying to understand implicit 3D trigonometry, specifically with this equation: $$\sin(y)+\cos(z)=\cos(x)$$ Can someone please explain to me what is going on with this equation? I really can't ...
1
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1answer
107 views

Is there a typo in Calculus:Early Transcedentals?

I just finished doing my homework on Local Linear Approximations in 3-space (Ch.13.4). In one of the problems the answer I got is different from the answer key. Problem 39. We have a function ...
1
vote
2answers
85 views

Normal from multiple vectors

I have a several 3D vectors $X_{i}$ which lay approximately in a plane. Now I need to find a single vector, which is normal (as much as possible) to all of them. For two vectors, I can use a cross ...
3
votes
0answers
765 views

Three-dimensional vectors and force systems

Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below As shown, a system of cables suspends a crate weighing W = 350 . (Part C 1 figure) ...
3
votes
0answers
52 views

Scale-agnostic, differentiable, co-planarity measure

I am looking for an (almost everywhere) differentiable function $f(p_1,p_2,p_3,p_4)$ that given four points will give me a scale-agnostic measure for co-planarity. It is zero if the four points lie on ...
1
vote
1answer
163 views

Differentiable orthogonal 3D vector

Does anybody know a simple and differentiable function that converts a 3D vector u = (x, y, z) to another vector that is orthogonal to ...