Determine a vector perpendicular to the plane

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SUMMARY

The discussion focuses on determining a vector perpendicular to a plane defined by the points (1,0,0), (2,0,-1), and (1,4,3). The user successfully calculates two vectors from the first point and finds their cross product, resulting in the perpendicular vector (4i - 3j + 4k). For the area of the triangle formed by these points, the user correctly identifies that it is half the magnitude of the perpendicular vector, calculating it to be approximately 3.201. The discussion also touches on a separate question regarding the center and radius of a sphere, which remains unclear.

PREREQUISITES
  • Vector algebra, specifically cross product calculations
  • Understanding of geometric concepts related to planes and triangles
  • Knowledge of magnitude and area calculations in three-dimensional space
  • Familiarity with sphere equations and their geometric interpretations
NEXT STEPS
  • Study vector cross product properties and applications in geometry
  • Learn how to calculate the area of triangles in three-dimensional space
  • Explore the equations of spheres and their geometric significance
  • Investigate the relationship between vectors and planes in linear algebra
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Students studying geometry, particularly those focusing on vector mathematics and three-dimensional shapes, as well as educators looking for examples of vector applications in geometric contexts.

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Homework Statement


this problem has 2 parts

a- Determine a vector perpendicular to the plane containing the points 1,0,0 2,0,-1 1,4,3
b - determine the area of the triangle

Homework Equations





The Attempt at a Solution



a- i found the two vectors from the 1st point a = i - k, b = 4j + 3k
and i believe a X b would yield a perpindicular vector
a X b = 4i - 3j + 4k, is this correct


b - isn't the area of the triangle 1/2 the magnitude of the perpindicular vecor

.5(4^2 + 3^2 + 4^2)^1/2 = 3.201 is this correct

b - i don't really understand what the question is asking

c - center is (4,-1,-3)
r = (-1 + (8^2 + 2^2 + 6^2)/4)^1/2 = 5 is this correct
 
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Your answers to a and b are correct. What is the question for c?
 

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