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The cross product is a mathematical operation used in physics to determine the vector that is perpendicular to two other vectors. It is widely used in torque calculation, where the force applied to an object and its position vector are crucial.For two vectors \( \mathbf{a} \) and \( \mathbf{b} \) in three-dimensional space defined as \( \mathbf{a}=a_x\mathbf{i} + a_y\mathbf{j} + a_z\mathbf{k} \) and \( \mathbf{b}=b_x\mathbf{i} + b_y\mathbf{j} + b_z\mathbf{k} \) respectively, their cross product \( \mathbf{a} \times \mathbf{b} \) results in a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \) and has a direction given by the right-hand rule.

The magnitude of this vector is given by the area of the parallelogram that the vectors span, which is equivalent to \( |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta) \) where \( \theta \) is the angle between the vectors. In the context of our exercise, the cross product helps us calculate the torque exerted on a particle. However, since the given vectors lie in the xy-plane, their cross product solely has a k-component because only the z-axis is perpendicular to the plane of the two vectors.

The vector position, often represented as \( \vec{r} \) in physics, is a vector that points from the origin of a coordinate system to the location of a point in space. This vector is essential for describing the position of objects and is commonly used in conjunction with other vectors, like forces, to perform calculations such as torque.In our exercise, the vector position of a particle is given by \( \vec{r}=(\hat{\mathbf{i}}+3\hat{\mathbf{j}}) \mathbf{m} \), where \( \hat{\mathbf{i}} \) and \( \hat{\mathbf{j}} \) are the unit vectors along the x-axis and y-axis, respectively. This means the particle is 1 meter along the x-axis and 3 meters along the y-axis from the origin. When we need to find the position relative to another point—other than the origin—we subtract the coordinates of the point from the coordinates of the particle as seen in the steps for calculating the torque about the point (0,6).

This concept is fundamental in physics because it allows for the determination of an object's location in space, which is paramount for further analysis of the object's motion or the effects of forces on it.

Forces are a fundamental concept in physics, defining the interaction that causes an object to move or try to move, stop or try to stop. The way forces affect an object's motion depends on their magnitude and direction, and they are typically represented by vectors.In our exercise, the force vector is \( \vec{F}=(3\hat{\mathbf{i}}+2 \hat{\mathbf{j}}) \mathbf{N} \), indicating that a force of 3 Newtons is applied along the x-axis and 2 Newtons along the y-axis. Understanding the role of forces is crucial for solving problems in mechanics, as it serves as an input for calculating torque, momentum, and many other physical quantities.

The combination of the force vector with the position vector through cross product computation, as explained previously, allows us to understand how the force causes rotation about a point—the essence of torque. Being intuitive with forces and how they interact with objects via vectors lays the groundwork not just for problem-solving but for grasping the physical laws governing motion and dynamics.