Solution:

Given, A B=C and |A| |B|=|C|

We need to determine the relationship between A and B.

1. Parallel Vectors:

If A and B are parallel vectors, then the angle between them is 0° or 180°. In this case, the dot product of A and B will be equal to the product of their magnitudes.

A.B = |A| |B| cosθ

Since |A| |B|=|C|, we can write:

A.B = |C| cosθ

But A B=C, so we can write:

A.A + A.B = |A|^2 + |B|^2

|A|^2 + C^2 = |C|^2

Substituting A.B = |C| cosθ in the above equation, we get:

|A|^2 + |C|^2 - 2|A||C|cosθ = |C|^2

Simplifying the above equation, we get:

|A|^2 = |C|^2 - 2|A||C|cosθ

Since |A| is always positive, the right-hand side of the above equation is non-negative. So, we have:

|C|^2 - 2|A||C|cosθ ≥ 0

But |C| = |A| |B|, so we can write:

|A|^2 - 2|A||B|cosθ + |B|^2 ≥ 0

This is the condition for A and B to be parallel vectors. It is a quadratic equation in |A|, which has real roots if and only if the discriminant is non-negative.

2. Perpendicular Vectors:

If A and B are perpendicular vectors, then the angle between them is 90°. In this case, the dot product of A and B will be zero.

A.B = |A| |B| cosθ = 0

Since |A| is always positive, we must have cosθ = 0. This implies that A and B are perpendicular to each other.

3. Anti-Parallel Vectors:

If A and B are anti-parallel vectors, then the angle between them is 180°. In this case, the dot product of A and B will be equal to the negative product of their magnitudes.

A.B = |A| |B| cosθ = -|A| |B|

Since |A| |B|=|C|, we can write:

A.B = -|C|

But A B=C, so we can write:

A.A - A.B = |A|^2 - |B|^2

|A|^2 - C^2 = |C|^2

Substituting A.B = -|C| in the above equation, we get:

|A|^2 + |C|^2 = |C|^2

This implies that |A|^2 = 0, which implies that A is a null vector.

4. Null Vector:

If A is a null vector, then its magnitude is zero. This implies that |C|=0, which in turn implies that B is also a null vector.