You slip a wrench over a bolt. Taking the origin at the bolt, the other end of the wrench is at \(x=18 \mathrm{cm}, y=5.5 \mathrm{cm} .\) You apply a force \(\vec{F}=88 \hat{\imath}-23 \hat{\jmath} \mathrm{N}\) to the end of the wrench. What's the torque on the bolt?

Understanding the cross product of vectors is essential in physics, especially when dealing with rotational motion. The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in another vector that is perpendicular to the plane containing the initial vectors.

The formula for the cross product of two vectors \( \vec{A} \times \vec{B} \) is given by \( \vec{C} \) where \( \vec{C} \) has a magnitude equal to \( |\vec{A}| |\vec{B}| \sin(\theta) \) and is directed perpendicularly to the plane formed by \( \vec{A} \) and \( \vec{B} \), according to the right-hand rule. In the context of torque calculation, you can use a simplified 2D cross product when forces act on a plane, as presented in the wrench problem above.

Torque is a fundamental concept in the study of rotational motion. It can be thought of as the rotational equivalent of force. While force causes an object to move or accelerate in a straight line, torque causes an object to rotate. The magnitude of the torque depends on two factors: the magnitude of the force applied and the distance from the axis of rotation to the point where the force is applied, known as the lever arm.

Mathematically, torque \( \vec{\tau} \) is the cross product of the position vector \( \vec{r} \) and the force vector \( \vec{F} \), represented by \( \vec{\tau} = \vec{r} \times \vec{F} \). A larger force or a longer lever arm results in greater torque. In the wrench problem, by multiplying the force by the distance from the bolt, we find the torque exerted on the bolt.

Solving physics problems often involves a systematic approach of identifying knowns, defining vector qualities, applying physics principles, and executing mathematical calculations. Step by step problem-solving helps to break down complex problems into manageable parts.

In the case of the wrench and bolt scenario, the process started with identifying the force and radius vectors, converting them into compatible units, and then using the cross product to find the torque. This methodical approach ensures each physical quantity is accounted for, allowing for an accurate solution. Clear identification and stepwise calculations are essential, particularly in introductory physics, to build a strong foundation for more complex problem-solving.

Vectors play a critical role in expressing quantities that have both magnitude and direction in physics. They are widely used to represent forces, velocities, accelerations, and other vector quantities. Vector mathematics allows one to perform operations like addition, subtraction, and multiplication in the form of dot and cross products.

In physics problems, it is crucial to understand how to handle vectors both graphically and numerically. In the wrench example, both the force applied and the position of the wrench are vector quantities, and the cross product of these yields the vector torque. Mastery of vector mathematics is integral to solving problems across various physics domains, from kinematics to electromagnetism.