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The cross product, often called the vector product, is a fundamental operation in vector algebra. It is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to both of the input vectors. The cross product is only defined in three-dimensional space. Its formula uses the components of the vectors, allowing us to compute the resultant perpendicular vector.
To compute the cross product between two vectors, \(\vec{a}= (a_1, a_2, a_3)\) and \(\vec{b}= (b_1, b_2, b_3)\), use the formula: \[(a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\]

• Useful for determining the area of a parallelogram formed by the two vectors.
• Commonly used in physics and engineering to find torques or rotational forces.
• Produces a vector orthogonal to the plane containing \(\vec{a}\) and \(\vec{b}\).

In the provided exercise, we see the cross product application in calculating \(\vec{p} \times \vec{q}\) to further compute with vector \(\vec{r}\). This operation is pivotal in expressing the vector relations between these sets.

The dot product, or scalar product, is another crucial operation in vector mathematics. It involves multiplying two vectors and results in a scalar quantity, often representing metrics like work done by a force. The dot product gives a sense of the component of one vector in the direction of another.
To find the dot product between two vectors, \(\vec{a}= (a_1, a_2, a_3)\) and \(\vec{b}= (b_1, b_2, b_3)\), use the formula: \[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]

• Effective in determining the angle between vectors. If the dot product is zero, the vectors are orthogonal.
• Used in computing projections of vectors.
• In physics, it is associated with quantities like work: \(\text{Work} = \text{Force} \cdot \text{Displacement}\).

In the context of the exercise, the dot products \(\vec{p} \cdot \vec{r}\) and \(\vec{q} \cdot \vec{r}\) are essential steps needed for expressing \( (\vec{p} \times \vec{q}) \times \vec{r} \) in terms of \(\vec{p}\) and \(\vec{q}\).

A linear combination involves taking multiple vectors and scaling them by coefficients, then summing the results. It is a fundamental concept in linear algebra, as it forms the basis for defining vector spaces and span. Any vector that can be described as a linear combination of other vectors lies in the span of those vectors.
Consider vectors \(\vec{a}\) and \(\vec{b}\), a linear combination is given by \(c_1\vec{a}+c_2\vec{b}\), where \(c_1\) and \(c_2\) are scalars.

• Represents vector equations and solutions in terms of others.
• Integral to forming expressions that describe physical situations, like those seen in linear dynamics.
• Helps in determining bases of vector spaces, part of simplifying complex calculations.

In the exercise, the linear combination is used to verify that \((\vec{p} \times \vec{q}) \times \vec{r}\) can be reconstructed using \(\vec{p}\) and \(\vec{q}\) in the form \((\vec{p} \cdot \vec{r}) \vec{q} - (\vec{q} \cdot \vec{r}) \vec{p}\). This illustrates the importance of expressing vector transformations in terms of familiar components.