Show that if \(a\) and \(b\) are parallel vectors then their vector product is the zero vector.

When we talk about parallel vectors, we're referring to a pair of vectors that have the same or opposite direction. In mathematical terms, two vectors are parallel if one is a scalar multiple of the other—this means you can multiply one vector by a single number (a scalar) to get the other vector. For example, if we have vector \( \mathbf{a} \) and vector \( \mathbf{b} \), and \( \mathbf{a} = k\mathbf{b} \), for some scalar \( k \), then \( \mathbf{a} \) and \( \mathbf{b} \) are parallel. It's a key concept when considering the cross product because if two vectors are parallel, their cross product will yield a particular outcome, which we explore in the context of the zero vector.

The cross product, also known as the vector product, is an operation that combines two vectors in three-dimensional space to produce a third vector that is perpendicular to the plane formed by the original vectors. This third vector's direction is determined by the right-hand rule, and its magnitude is related to the area of the parallelogram that the vectors span. Mathematically, the cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \times \mathbf{b} = \Vert \mathbf{a} \Vert \Vert \mathbf{b} \Vert \sin{\theta} \hat{\mathbf{n}} \), where \( \Vert \mathbf{a} \Vert \) and \( \Vert \mathbf{b} \Vert \) represent the magnitudes of the vectors, \( \theta \) is the angle between them, and \( \hat{\mathbf{n}} \) is the unit vector perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This definition is crucial as it sets up the relationship between the vectors and their resulting cross product.

Understanding the magnitude of a vector is a fundamental aspect of vector operations. The magnitude of a vector is essentially its 'length' and is calculated by treating the vector as the hypotenuse of a right triangle, with its components acting as the other sides. For a 3-dimensional vector \( \mathbf{v} = (v_x, v_y, v_z) \), the magnitude is determined by the formula \( \Vert \mathbf{v} \Vert = \sqrt{v_x^2 + v_y^2 + v_z^2} \). This calculation gives a scalar value, representing how far the vector extends in space. When discussing the cross product, the magnitudes of the vectors involved affect the magnitude of the resulting vector.

The zero vector, denoted by \( \mathbf{0} \), plays a unique role in vector mathematics. It is a vector of zero magnitude and has no specific direction. In the context of the cross product, when the cross product of two vectors \( \mathbf{a} \times \mathbf{b} \) results in the zero vector, this means that \( \mathbf{a} \) and \( \mathbf{b} \) are either parallel or one of them is the zero vector to begin with. As seen in the exercise, when two parallel vectors are crossed, since \( \sin{\theta} = 0 \) (with \( \theta \) being either 0 or \( \pi \)), the resulting vector's magnitude is zero, leading to the conclusion that the cross product is the zero vector, symbolizing an intersection of concepts central to understanding vector operations.