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In physics, the concept of a displacement vector is crucial to describe how an object's position changes over time. Displacement is different from distance because it considers the direction of movement, not just the magnitude. To calculate the displacement vector, one subtracts the initial position vector from the final position vector.
For instance, if the initial position vector is given as \( \vec{r}_{\text{initial}} = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \) and the final position vector as \( \vec{r}_{\text{final}} = -2.0 \hat{\mathrm{i}} + 8.0 \hat{\mathrm{j}} - 2.0 \hat{\mathrm{k}} \), the displacement vector \( \Delta \vec{r} \) is calculated by:

• Taking the difference in the i-components: \( -2.0 - 5.0 = -7.0 \)
• Taking the difference in the j-components: \( 8.0 - (-6.0) = 14.0 \)
• Taking the difference in the k-components: \( -2.0 - 2.0 = -4.0 \)

This yields the displacement vector \( \Delta \vec{r} = -7.0 \hat{\mathrm{i}} + 14.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \), which provides a clear representation of the change in position in each direction.

Unit vector notation is a streamlined way to denote vectors, often used in physics to express vectors in three-dimensional space. A unit vector has a magnitude of one and indicates direction along one of the axes, specifically represented by \( \hat{\mathrm{i}} \), \( \hat{\mathrm{j}} \), and \( \hat{\mathrm{k}} \). Each of these unit vectors represents unit displacement along the x, y, or z axes, respectively.
When writing vectors in unit vector notation, like \( 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \), the coefficients of the unit vectors (5.0, -6.0, and 2.0) represent the vector's components along each axis. This notation makes it easy to perform operations such as addition and subtraction on vectors by focusing on each component individually. Furthermore, unit vector notation is compact and conveniently handles the mathematical rigor required in physics and engineering applications.
It's important to remember that while a unit vector directs along an axis, it does not imply any specific distance, serving solely as a directional guide.

Vector subtraction is a fundamental operation needed to find differences between vectors, as seen in displacement calculations. In vector subtraction, each corresponding component of the second vector is subtracted from the first vector. For example, to find the displacement vector, we align the initial and final position vectors and operate component-wise.
Consider vectors \( \vec{A} = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \) and \( \vec{B} = -2.0 \hat{\mathrm{i}} + 8.0 \hat{\mathrm{j}} - 2.0 \hat{\mathrm{k}} \). To compute \( \vec{B} - \vec{A} \), subtract the components:

• i-component: \( -2.0 - 5.0 = -7.0 \)
• j-component: \( 8.0 - (-6.0) = 14.0 \)
• k-component: \( -2.0 - 2.0 = -4.0 \)

This yields \( \Delta \vec{r} = -7.0 \hat{\mathrm{i}} + 14.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \).
Keep in mind that vectors, unlike scalars, account for direction, making subtraction not only a simple arithmetic operation but also a way to identify precise vectors resulting from position change.