Study guide and
10 practice problems
on:
Lines and planes
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Study Guide
The 2d line going through $\bfx_0$ with normal vector $\bfn$ is given by $$\bfn \cdot \bfx = \bfn \cdot \bfx_0$$
(4 problems)
The plane going through $\bfx_0$ with normal vector $\bfn$ is given by $\bfn \cdot \bfx = \bfn \cdot \bfx_0.$
(4 problems)
The line going through $\bfx_0$ with tangent vector $\bfv$ is given by $\bfx(t) = \bfx_0 + t \bfv$.
(1 problem)
Intersection of two planes is typically a line
(0 problems)
Intersection of a plane with a line is typically a point
(0 problems)
Related topics
Multivariable calculus
(147 problems)
Lines, Planes, and Curves
(13 problems)
Practice problems
A frisbee of radius $r$ translates rightward at speed $v$ meter/second. It rotates clockwise at $\omega$ radian/second. Initially the frisbee is centered at the origin.
Find the trajectory swept by the point initially at $(0,r)$.
Compute the speed as a function of time.
Describe when the speed is largest and smallest.
Solution
Find the line containing the points $(0, 1)$ and $(3, 0)$ by computing a normal vector to the line. Write it in the form $\bfn \cdot \bfx = b$.
Solution
Find a normal vector to the 2d line $2x - y = 5$.
Solution
Sketch the region in 2d satisfied by $ ( 1, 1 ) \cdot \bfx \geq 2.$
Solution
Consider the set of 3d points whose component (scalar projection) along the vector $\bfv$ is the constant $c$. Show that these points form a plane, and find that plane.
Solution
Find a normal vector to the plane $2x + y - 3z = 1$.
Solution
Find the plane containing the points $(a,0,0)$, $(0, b, 0)$, and $(0,0,c)$.
Solution
Find the tangent plane at $(1,1,1)$ to the surface $$x^2 + y^2 + z^2 + xy + xz = 5.$$
Solution
Give the equation for the tangent plane to the surface $z x^2 + x y^2 + y z^2 = 5$ at the point $(-1,1,2)$.
Solution
Find a normal vector to the curve $\sqrt{x} + \sqrt{y} = 2$ at $(x,y) = (1,2)$. Use it to find the tangent line at $(1,1)$ expressed in the form $\bfn \cdot \bfx = b$.
Solution