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The cross product is a mathematical operation used extensively in physics to find quantities like torque. It takes two vectors as input and produces another vector that is perpendicular to both input vectors. For vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), the cross product \( \vec{A} \times \vec{B} \) is defined as:

• \( (A_yB_z - A_zB_y) \hat{i} \)
• \( (A_zB_x - A_xB_z) \hat{j} \)
• \( (A_xB_y - A_yB_x) \hat{k} \)

This gives us a new vector. In this exercise, torque \( \vec{\tau} \) is calculated using the cross product \( \vec{\tau} = \vec{r} \times \vec{F} \), where \( \vec{r} \) is the position vector and \( \vec{F} \) is the force. The resulting torque vector is perpendicular to the plane containing \( \vec{r} \) and \( \vec{F} \), pointing in the direction given by the right-hand rule.

The dot product is another way of multiplying two vectors, but instead of producing a vector, it results in a scalar quantity. It's particularly useful for finding the angle between two vectors and understanding how much one vector lies in the direction of another. For vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} \), the dot product is calculated as:

• \( \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y \)

This operation summarizes how much of vector \( \vec{A} \) is pointing in the same direction as vector \( \vec{B} \). In the context of the exercise, the dot product formula \( \vec{r} \cdot \vec{F} = |\vec{r}| |\vec{F}| \cos \theta \) reveals that if the result is zero, the vectors are perpendicular. Thus, knowing the dot product helps us find critical information like angles between vectors.

Understanding the angle between vectors is crucial in many physical and mathematical applications. The angle \( \theta \) between two vectors \( \vec{A} \) and \( \vec{B} \) can be found using the dot product, with the formula:\[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]If the dot product \( \vec{A} \cdot \vec{B} \) is zero, it means the two vectors are perpendicular, making \( \theta = 90^\circ \). In this exercise, we see this occurs with vectors \( \vec{r} \) and \( \vec{F} \), as their dot product is zero, confirming they are at a right angle to each other. This characteristic is essential not only in calculating torque but also in identifying orthogonal directions.

The magnitude of a vector provides the length or size of the vector. It is a crucial concept when dealing with physical quantities like force, velocity, or displacement. For a vector \( \vec{A} = A_x \hat{i} + A_y \hat{j} \), the magnitude \( |\vec{A}| \) is calculated by:\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]The magnitude gives us a sense of the vector's strength without regard to its direction. For example, in the problem, the magnitude of the position vector \( \vec{r} \) is 5 meters, while that of the force vector \( \vec{F} \) is 10 Newtons. These magnitudes help calculate the dot product and understand the scale of the physical entities represented by the vectors.