Understanding the Angle Between Vectors \(\vec{A}+\vec{B}\) and \(\vec{A}\times\vec{B}\)

The question asks for the angle between vectors \(\vec{A}+\vec{B}\) and \(\vec{A}\times\vec{B}\), given two vectors \(\vec{A}\) and \(\vec{B}\).

Analyzing Vector Addition (\(\vec{A}+\vec{B}\))

The vector sum \(\vec{A}+\vec{B}\) is obtained by adding vectors \(\vec{A}\) and \(\vec{B}\). Geometrically, using the parallelogram or triangle law of Vector Addition, the resultant vector \(\vec{A}+\vec{B}\) lies in the same plane as vectors \(\vec{A}\) and \(\vec{B}\). If \(\vec{A}\) and \(\vec{B}\) are non-collinear, they define a unique plane. The vector \(\vec{A}+\vec{B}\) will always lie within this plane or be collinear with \(\vec{A}\) or \(\vec{B}\) if they are parallel.

This property of Vector Addition is fundamental to understanding the spatial orientation of the resultant vector.

Analyzing the Cross Product (\(\vec{A}\times\vec{B}\))

The Cross Product \(\vec{A}\times\vec{B}\) is a vector operation between \(\vec{A}\) and \(\vec{B}\) that results in a new vector. A key Vector Property of the cross product is its direction. The resulting vector \(\vec{A}\times\vec{B}\) is always perpendicular to both vector \(\vec{A}\) and vector \(\vec{B}\). This means \(\vec{A}\times\vec{B}\) is perpendicular to the plane containing both \(\vec{A}\) and \(\vec{B}\).

The direction is determined by the right-hand rule, but the crucial aspect for finding the Angle Between Vectors here is its perpendicularity to the plane defined by \(\vec{A}\) and \(\vec{B}\).

Determining the Angle Between the Vectors

We have established the following:

• Vector \(\vec{A}+\vec{B}\) lies in the plane containing \(\vec{A}\) and \(\vec{B}\).
• Vector \(\vec{A}\times\vec{B}\) is Perpendicular Vectors to the plane containing \(\vec{A}\) and \(\vec{B}\).

Therefore, vector \(\vec{A}+\vec{B}\) and vector \(\vec{A}\times\vec{B}\) are Perpendicular Vectors to each other, provided \(\vec{A}+\vec{B}\) is not the zero vector (which happens only if \(\vec{A} = -\vec{B}\) and both are non-zero, or if both are zero vectors) and \(\vec{A}\times\vec{B}\) is not the zero vector (which happens if \(\vec{A}\) and \(\vec{B}\) are parallel or one of them is a zero vector). However, the question assumes general vectors where these products exist as non-zero vectors.

The angle between any vector lying in a plane and a vector perpendicular to that plane is always 90 degrees. This is a fundamental geometric and Vector Property.

Thus, the angle between vectors \(\vec{A}+\vec{B}\) and \(\vec{A}\times\vec{B}\) is 90°.

Conclusion

Based on the directional properties of vector addition and the Cross Product, the vector sum lies in the plane of the original vectors, while the cross product is perpendicular to that plane. Consequently, the angle between the resultant vector from Vector Addition and the vector from the cross product is 90 degrees.

The final answer is indeed 90°, representing the specific Angle Between Vectors with these properties.