Show that \(\vec{A} \cdot(\vec{A} \times \vec{B})=0\) for any vectors \(\vec{A}\) and \(\vec{B}\).

The dot product, also known as the scalar product, is one of the fundamental operations you can perform with vectors. Imagine you have two vectors, \(\vec{A}\) and \(\vec{B}\), and you want to find out how much one vector extends in the direction of the other. The dot product gives you the answer in the form of a single number, a scalar.

Mathematically, if \(\vec{A} = [A_x, A_y, A_z]\) and \(\vec{B} = [B_x, B_y, B_z]\), the dot product is defined as \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\). It has significant implications in physics and geometry, one of which is determining whether two vectors are orthogonal. Two vectors are orthogonal if their dot product is zero. This concept is crucial in solving many types of geometric problems.

• The dot product is commutative, which means \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\).
• It is a measure of the 'projection' of one vector onto another.
• The dot product of two orthogonal vectors is always zero.

The cross product, or vector product, is a binary operation on two vectors in three-dimensional space and is denoted by \(\times\). Unlike the dot product, the result of a cross product is not a scalar but a vector that is orthogonal to the plane formed by the two input vectors.

For vectors \(\vec{A}\) and \(\vec{B}\), expressed as \(\vec{A} = [A_x, A_y, A_z]\) and \(\vec{B} = [B_x, B_y, B_z]\), the cross product is \(\vec{A} \times \vec{B} = [A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x]\).

This new vector's magnitude is proportional to the area of the parallelogram that \(\vec{A}\) and \(\vec{B}\) span, providing a measure of the vectors' 'mutual perpendicularity'.

• The cross product is anti-commutative, meaning \(\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})\).
• It is useful for finding normal vectors for planes and in computing torques in physics.
• Cross product magnitude can be calculated using the sine of the angle between the vectors and their magnitudes.

Vectors are said to be orthogonal to each other if they meet at a right angle (90 degrees). In three-dimensional space, the concept of orthogonality is essential because it helps to define directions that are independent of one another. For instance, the 'up' direction is orthogonal to the 'north' and 'east' directions.

If two vectors are orthogonal, their dot product is zero, as seen in our original exercise where \(\vec{A}\) dot \(\vec{A} \times \vec{B} = 0\). This property is a quick way to check if two vectors are at right angles without having to measure the angle directly.

• Orthogonal vectors have a zero dot product.
• In coordinate systems, axes are typically defined by unit vectors that are orthogonal to each other.
• Orthogonality is used in vector projections, to extract independent components of motion, and in many areas like computer graphics and calculus.