Area of the triangle is measure of the area covered by the triangle. The area of the triangle is half of the magnitude. The area of the triangle is 8.15 squared units.
Given information-
A triangle has vertices (0, 0, 0), (1, 1, 1), and (0, −4, 9).
Compute the vector from (0, 0, 0), (1, 1, 1),
[tex]\vec a=(0-1) \hat i+(0-1) \hat j+(0-1) \hat k\\ \vec a=- \hat i- \hat j-\hat k\\[/tex]
Compute the vector from (0, 0, 0), (1, -4,9),
[tex]\vec b=(0-0) \hat i+(0-(-4)) \hat j+(0-9) \hat k\\ \vec b=4 \hat j-9 \hat k[/tex]
Area of the triangle is measure of the area covered by the triangle. The area of the triangle is half of the magnitude.
Cross product of the two vectors,
[tex]\vec c=\vec a \times \vec b\\\vec c=(- \hat i- \hat j-\hat k)\times(\hat 4j-9\hat k)\\\vec c=13 \hat i- 9\hat j+4\hat k[/tex]
Magnitude is the cross product of the two vectors,
[tex]|\vec c|=\sqrt{13^2+9^2+4^2} \\ |\vec c|=\sqrt{169+81+16} \\ |\vec c|=\sqrt{266} \\ |\vec c|=16.31[/tex]
Thus the area of the triangle is,
[tex]A=\dfrac{16.31}{2} \\ A=8.15[/tex]
Thus the area of the triangle is 8.15 squared units.
Learn more about the area of the triangle here;
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