find the area of the parallelogram that has vectors as adjacent sides
u = -2i +j +5k
v = 4i -3j -3k
To find the area of a parallelogram using vectors, you can use the cross product of the two adjacent sides. The magnitude of the cross product gives you the area of the parallelogram.
1. Start by calculating the cross product of the two vectors u and v.
- Take the determinant of the 3x3 matrix formed by the coefficients of i, j, and k:
u x v = |-2 1 5|
| 4 -3 -3|
- Calculate the determinants of the 2x2 submatrices:
u x v = (-2 × -3 - 1 × -3)i - (-2 × -3 - 5 × 4)j + (-2 × -3 - 1 × 4)k
= (6 + 3)i - (6 - 20)j + (6 + 4)k
= 9i + 14j + 10k
2. Find the magnitude of the cross product by taking the square root of the sum of the squares of its components:
|u x v| = √(9² + 14² + 10²)
= √(81 + 196 + 100)
= √377
3. The magnitude of the cross product represents the area of the parallelogram formed by the adjacent sides u and v:
Area = √377
Therefore, the area of the parallelogram with vectors u = -2i + j + 5k and v = 4i - 3j - 3k is √377.
recall area of parallelogram using vectors u and v
= | u X v|
go for it, you have u = (-2,1,5) and v = (4,-3,-3)