Explanation
1. Define the Position Vectors
Let the vertices of the triangle be A, B, and C. Their position vectors are:
2. Find the Side Vectors
To find the area, we first need two vectors representing two sides of the triangle, such as AB and AC.
AB=b−a=(5−3)i^+(2−1)j^+(1−0)k^=2i^+j^+k^
AC=c−a=(1−3)i^+(−2−1)j^+(3−0)k^=−2i^−3j^+3k^
3. Calculate the Cross Product
The area of the triangle is given by the formula: Area=21∣AB×AC∣.
AB×AC=i^2−2amp;j^amp;1amp;−3amp;k^amp;1amp;3
=i^(3−(−3))−j^(6−(−2))+k^(−6−(−2))
4. Find the Magnitude and Final Area
Now, calculate the magnitude of the cross product:
Finally, the area of the triangle is:
Area=21×229=29 sq. units
Correct Option: D) 29
