Find the area of a parallelogram whose adjacent sides are given by the vectors a = 3i + 4k and b = 2i - 3j + 5k.

To find the area of a parallelogram formed by two vectors, we can use the cross product of those vectors. The area is equal to the magnitude of the cross product.

Step 1: Define the Vectors

We have the following vectors:

• Vector a: 3i + 0j + 4k
• Vector b: 2i - 3j + 5k

Step 2: Calculate the Cross Product

The cross product of vectors a and b can be calculated using the determinant of a matrix:

|   i    j    k   |
|   3    0    4   |
|   2   -3    5   |

Calculating the determinant, we find:

i(0*5 - 4*(-3)) - j(3*5 - 4*2) + k(3*(-3) - 0*2)

This simplifies to:

i(0 + 12) - j(15 - 8) + k(-9)

So, the cross product is:

12i - 7j - 9k

Step 3: Find the Magnitude

The magnitude of the cross product gives us the area:

Area = √(12² + (-7)² + (-9)²)

Calculating this:

Area = √(144 + 49 + 81) = √274

Final Result

The area of the parallelogram is:

√274 square units