If two vector , A vector = 1i+6j-5k , B vector = 2i+3j+4k represents the two adjecents sides of a parallelogram , find the area of the parallelogram .
the area of the parallelogram is just |A×B| = |39i-14j-9k| = √1798
To find the area of the parallelogram formed by two vectors, we can use the cross product.
Given vectors:
A = 1i + 6j - 5k
B = 2i + 3j + 4k
1. Calculate the cross product of the two vectors:
Cross product, A x B = (A_y * B_z - A_z * B_y)i + (A_z * B_x - A_x * B_z)j + (A_x * B_y - A_y * B_x)k
Substituting the values:
A x B = (6 * 4 - (-5) * 3)i + ((-5) * 2 - 1 * 4)j + (1 * 3 - 6 * 2)k
= (24 + 15)i + (-10 - 4)j + (3 - 12)k
= 39i - 14j - 9k
2. Calculate the magnitude of the cross product:
|A x B| = √[(39)^2 + (-14)^2 + (-9)^2]
= √(1521 + 196 + 81)
= √1798
≈ 42.42
3. The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors. Therefore, the area of the parallelogram is approximately 42.42 square units.
To find the area of a parallelogram given its adjacent sides, you can use the cross product of the two vectors representing the adjacent sides.
Step 1: Calculate the cross product of the two vectors A and B.
The cross product of two vectors A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃) is given by:
A x B = (A₂B₃ - A₃B₂)i - (A₁B₃ - A₃B₁)j + (A₁B₂ - A₂B₁)k
In this case, A = (1, 6, -5) and B = (2, 3, 4). Plugging in these values, the cross product A x B is:
A x B = (6 × 4 - (-5) × 3)i - (1 × 4 - (-5) × 2)j + (1 × 3 - 6 × 2)k
= 47i - 14j - 9k
Step 2: Calculate the magnitude of the cross product.
The magnitude of a vector v = (v₁, v₂, v₃) is given by:
|v| = √(v₁² + v₂² + v₃²)
In this case, |A x B| = √(47² + (-14)² + (-9)²)
= √(2209 + 196 + 81)
= √2486 ≈ 49.86
Step 3: Calculate the area of the parallelogram.
The area of a parallelogram formed by adjacent sides A and B is equal to the magnitude of their cross product.
Area = |A x B| ≈ 49.86
Therefore, the area of the parallelogram formed by the vectors A = 1i + 6j - 5k and B = 2i + 3j + 4k is approximately 49.86.