A.B×A.A = ABcos(theta)× AAcos(theta) = ABcos(theta) × AA = (A^3)B cos(theta)sin(theta)

Angle between vectors A and B:
The angle between two vectors can be determined using the dot product formula:

A · B = |A| |B| cos(theta)

where A · B is the dot product of vectors A and B, |A| and |B| are the magnitudes of vectors A and B respectively, and theta is the angle between the vectors.

Triple product A · (B × A):
The triple product A · (B × A) involves the dot product of vector A with the cross product of vectors B and A. Let's break down the expression step by step:

1. Compute the cross product (B × A):
The cross product of two vectors, B and A, is given by the formula:

B × A = |B| |A| sin(theta) n

where |B| and |A| are the magnitudes of vectors B and A respectively, theta is the angle between the vectors, and n is the unit vector perpendicular to the plane formed by B and A.

2. Take the dot product with vector A:
Now, we take the dot product of vector A with the cross product (B × A):

A · (B × A) = A · (|B| |A| sin(theta) n)

Using the distributive property of dot product, we get:

A · (B × A) = |B| |A| sin(theta) (A · n)

3. Simplify the dot product (A · n):
The dot product of vector A with the unit vector n can be simplified as:

A · n = |A| cos(phi)

where |A| is the magnitude of vector A and phi is the angle between vector A and the unit vector n.

4. Substitute the simplification:
Substituting the simplification into the previous equation, we have:

A · (B × A) = |B| |A| sin(theta) |A| cos(phi)

Using the trigonometric identity sin(theta) cos(phi) = 1/2 sin(2theta), we can further simplify the expression:

A · (B × A) = 1/2 |B| |A|^2 sin(2theta)

Conclusion:
The value of the triple product A · (B × A) is 1/2 |B| |A|^2 sin(2theta). The triple product involves the magnitudes of vectors A and B, as well as the angle theta between them.