The two adjacent sides of a parallelogram are 2i - 4j + 5k and 2i - 2j - 3k. Find the unit vector parallel to its diagonal. Also, find its area.

To find the unit vector parallel to the diagonal of a parallelogram formed by the vectors **A** = 2i - 4j + 5k and **B** = 2i - 2j - 3k, we first need to calculate the diagonal vector. The diagonal can be represented as **A** + **B**.

Step 1: Calculate the Diagonal Vector

Adding the two vectors:

• **A** + **B** = (2i - 4j + 5k) + (2i - 2j - 3k)
• Combine like terms: (2 + 2)i + (-4 - 2)j + (5 - 3)k
• This results in: 4i - 6j + 2k

Step 2: Find the Magnitude of the Diagonal Vector

The magnitude of the diagonal vector **D** = 4i - 6j + 2k is calculated using the formula:

|**D**| = √(4² + (-6)² + 2²) = √(16 + 36 + 4) = √56 = 2√14.

Step 3: Determine the Unit Vector

The unit vector **U** in the direction of the diagonal is given by:

**U** = **D** / |**D**| = (4i - 6j + 2k) / (2√14).

Thus, the unit vector is:

• **U** = (2/√14)i - (3/√14)j + (1/√14)k.

Step 4: Calculate the Area of the Parallelogram

The area **A** of the parallelogram can be found using the cross product of vectors **A** and **B**:

**A** × **B** = |**A**||**B**|sin(θ), where θ is the angle between the vectors.

Cross Product Calculation

Using the determinant method:

• **A** × **B** = |i j k|
• |2 -4 5|
• |2 -2 -3|

Calculating the determinant gives:

• i((-4)(-3) - (5)(-2)) - j((2)(-3) - (5)(2)) + k((2)(-2) - (-4)(2))
• i(12 + 10) - j(-6 - 10) + k(-4 + 8)
• = 22i + 16j + 4k.

Magnitude of the Cross Product

Now, find the magnitude:

|**A** × **B**| = √(22² + 16² + 4²) = √(484 + 256 + 16) = √756 = 6√21.

Final Results

The unit vector parallel to the diagonal is:

**U** = (2/√14)i - (3/√14)j + (1/√14)k.

The area of the parallelogram is:

Area = 6√21 square units.