91Ó°ÊÓ

In physics, we often use unit vector notation to express vectors in a concise and easy-to-interpret form. A vector can be expressed in terms of its components along the x, y, and z axes using the unit vectors \( \hat{i}, \hat{j}, \text{ and } \hat{k} \). Each unit vector has a magnitude of one and points in the direction of the corresponding axis. For example, a vector \( \vec{v} \) in unit vector notation might look like \( 3.0 \hat{i} - 2.0 \hat{j} + 4.0 \hat{k} \).
This system allows us to easily perform vector operations like addition, subtraction, and the calculation of torque, as demonstrated in the original exercise. By breaking vectors down into these components, we can systematically analyze their behavior in physical systems. Each component acts independently along its respective axis, simplifying many calculations.

The cross product is a mathematical operation that allows us to find the torque produced by a force acting on an object. Torque is a measure of the rotational effect a force has when applied to an object at some distance from a pivot point.
The cross product of two vectors \( \vec{A} \text{ and } \vec{B} \) in three dimensions is denoted by \( \vec{A} \times \vec{B} \). It results in a new vector that is perpendicular to both \( \vec{A} \) and \( \vec{B} \). This property is crucial in mechanics, where torque \( \vec{\tau} \) is determined using the cross product of the position vector \( \vec{r} \) and the force vector \( \vec{F} \):

• \( \vec{\tau} = \vec{r} \times \vec{F} \)

The magnitude of the cross product is given by \( |\vec{A}| |\vec{B}| \sin(\theta) \), where \( \theta \) is the angle between the vectors. In practical computations, such as those in the exercise, we use determinants to solve for torque in unit vector notation. This provides us with the torque vector components along the \( \hat{i}, \hat{j}, \text{ and } \hat{k} \) directions.

A position vector describes the location of a point in three-dimensional space relative to an origin. In the context of torque calculation, the position vector \( \vec{r} \) represents the distance and direction from the pivot point (usually the origin) to the point of force application.
For example, in the given exercise, the jar's position vector is represented as:

• \( \vec{r} = 3.0 \hat{i} - 2.0 \hat{j} + 4.0 \hat{k} \text{ m} \)

This vector indicates that the jar is 3 meters along the x-axis, -2 meters along the y-axis, and 4 meters along the z-axis. Knowing the position vector is essential for calculating torque using the cross product, as it provides the necessary information about where the force is applied relative to the pivot.
Adjusting the reference point for torque involves changing the position vector, as demonstrated in the modified torque calculation about a new point.

The vector sum, or resultant vector, is the vector that results from adding two or more vectors together. This is a key concept in physics as it helps us determine the net effect of multiple forces acting on a body.
In the exercise, two force vectors \( \vec{F}_1 \text{ and } \vec{F}_2 \) are summed to find their combined effect. Vector addition follows the rule of summing up corresponding components along each axis:

• \( \vec{F}_1 + \vec{F}_2 = (3.0 - 3.0) \hat{i} + (-4.0 - 4.0) \hat{j} + (5.0 - 5.0) \hat{k} = 0 \hat{i} - 8 \hat{j} + 0 \hat{k} \text{ N} \)

According to these calculations, the resultant force acts only along the negative y-axis. The concept of vector sum is vital for determining the net force that influences an object's motion or equilibrium and is particularly useful in fields like engineering and physics.