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Mathematics-Online problems:

Interactive Problem 122: Plane, Distance Point/Plane, Area of a Triangle

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the following plane in $ \mathbb{R}^3$

$\displaystyle E: \quad \vec{x}=\begin{pmatrix}-1 \\ 1 \\ 1 \end{pmatrix}+ \alp... ...\ 2 \\ 0 \end{pmatrix} + \beta \begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix} \; . $

a)
Let $ F$ be the plane through point $ A=(-4,2,2)$ , parallel to $ E$ . Find the equation describing the plane $ E$ .

b)
Which point $ B$ on plane $ E$ has minimal distance from point $ A$ . What is the minimal distance?

c)
Show that point $ C=(-3,0,4)$ lies in the plane $ F$ , and that the points $ A,B,C$ form an equilateral triangle. Find the lenghts of the sides, all interior angles, and the area of the triangle.

Solution:

a)
Complete the missing coefficients of the equation of $ F$ : $ 2x+$ $ y+$ $ z=$ .
b)
Point $ B=\Big($ , , $ \Big)$ , distance: .
c)
squared sides: $ \big\vert\overrightarrow{AB}\big\vert^2=$ , $ \big\vert\overrightarrow{BC}\big\vert^2=$ , $ \big\vert\overrightarrow{CA}\big\vert^2=$ .
$ \sphericalangle (ABC)=\pi/$ , $ \sphericalangle (BCA)=\pi/$ , $ \sphericalangle (CAB)=\pi/$ .
Area of triangle: /     (given as completely reduced fraction).

   

see also:

  automatically generated: 8/11/2017