Answer to Question #99172 in Analytic Geometry for nicol

Question

A parallelogram is formed in R3 by the vectors = (3, 2, –3) and = (4, 1, 5).
The point P = (0, 2, 3).
a. Determine the location of the vertices.
b. Determine the vectors representing the diagonals.
c. Determine the length of the diagonals.

Answer 1

Solution:a) Let PMNK - a parallelogram.

$$\overrightarrow {a}=\overrightarrow {PK}$$

$$\overrightarrow {PK} (x; y-2; z-3)$$ $$x=3; y-2=2; y=4; z-3=-3; z=0. K (3;4;0)$$

$$\overrightarrow {b}= \overrightarrow {PM}$$

$$\overrightarrow {PM} (x; y-2; z-3)$$

$$x=4; y-2=1; y=3; z-3=5; z=8. M(4;3;8)$$

$$\overrightarrow {a} = \overrightarrow {MN}$$

$$\overrightarrow {MN} (x-4; y-3; z-8)$$

$$x-4=3; x=7; y-3=2; y=5; z-8=-3; z=5. N(7;5;5)$$

b)

$$\overrightarrow {PN} (7-0;5-2;5-3)=\overrightarrow {PN} (7;3;2)$$

$$\overrightarrow {MK} (3-4;4-3;0-8)=\overrightarrow {MK} (-1;1;-8)$$

c)

$$\vert \overrightarrow {PN} \vert= \sqrt {49+9+4} = \sqrt {62}$$

$$\vert \overrightarrow {MK} \vert = \sqrt {1+1+64} = \sqrt {66}$$