Find the area of a triangle with vertices A(1, 3, 5), B(-2, -3, -4) and C(0, 3, -1).
To find the area of a triangle given the coordinates of its vertices, you can use the formula for the area of a triangle in three-dimensional space.
The formula is:
Area = 1/2 * |(B - A) x (C - A)|
Step 1: Find the vectors AB and AC
To find the vectors AB and AC, we subtract the coordinates of vertex A from the coordinates of vertices B and C respectively.
AB = B - A = (-2, -3, -4) - (1, 3, 5) = (-3, -6, -9)
AC = C - A = (0, 3, -1) - (1, 3, 5) = (-1, 0, -6)
Step 2: Find the cross product of AB and AC
To find the cross product, we perform a cross multiplication of the vectors AB and AC.
Cross product: AB x AC = (-3, -6, -9) x (-1, 0, -6)
To compute the cross product, we can calculate the determinants of the matrix formed by the coefficients of the vectors. The cross product is then given by the ordered triple of determinants.
Cross product: AB x AC = ((-6)*(-6) - (0)*(-9), (-9)*(-1) - (-6)*(-3), (-3)*0 - (-6)*(-1))
= (-36, -27, 0)
Step 3: Find the magnitude of the cross product
The magnitude of the cross product is the length of the vector AB x AC and can be found using the formula: Magnitude = √(x^2 + y^2 + z^2)
Magnitude of the cross product: |AB x AC| = √((-36)^2 + (-27)^2 + 0^2)
= √(1296 + 729)
= √2025
= 45
Step 4: Calculate the area of the triangle
Finally, we can calculate the area of the triangle using the formula: Area = 1/2 * |AB x AC|
Area = 1/2 * |AB x AC| = 1/2 * 45 = 22.5
Therefore, the area of the triangle with vertices A(1, 3, 5), B(-2, -3, -4), and C(0, 3, -1) is 22.5 square units.