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Vectors are fundamental components in physics and engineering that have both a direction and a magnitude. The magnitude of a vector is a measure of how long or large the vector is. If we have a two-dimensional vector represented as \(\vec{v} = v_x \hat{\mathrm{i}} + v_y \hat{\mathrm{j}}\), its magnitude \(|\vec{v}|\) is calculated using the Pythagorean theorem: \[|\vec{v}| = \sqrt{v_x^2 + v_y^2} .\] This formula helps us understand the size of the vector. For instance, considering vector \(\vec{a} = (4.0 \mathrm{~m}) \hat{\mathrm{i}} - (3.0 \mathrm{~m}) \hat{\mathrm{j}}\), we find its magnitude by substituting into the formula: - \(|\vec{a}| = \sqrt{4.0^2 + (-3.0)^2} = 5.0 \mathrm{~m}\). The magnitude tells us that vector \(\vec{a}\) has a length of \(5.0 \mathrm{~m}\) in the plane. Understanding vector magnitude is crucial for calculating distances in physics and engineering.

The angle of a vector indicates its direction relative to a reference direction, which is often the positive x-axis, represented by \(\hat{\mathrm{i}}\). To find the angle \(\theta\) of a vector relative to \(\hat{\mathrm{i}}\), we use the inverse tangent function: \[\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right) .\] This formula provides the angle in degrees or radians, depending on the system you are using. For example, for vector \(\vec{a} = 4.0 \hat{\mathrm{i}} - 3.0 \hat{\mathrm{j}}\): - The angle \(\theta\) is calculated as \(\theta = \tan^{-1}\left(\frac{-3.0}{4.0}\right)\). - This results in \(\theta \approx -36.87^\circ\). The negative sign indicates that the vector points below the positive x-axis. Knowing a vector's angle is crucial for determining how it points in space, which is vital in applications such as navigation and rotational mechanics.

Vector subtraction is used to find the difference between two vectors, and it is a critical operation in physics and engineering for analyzing relative motion or change in velocities. The process is straightforward: given two vectors \(\vec{a}\) and \(\vec{b}\), their difference is a vector \(\vec{a} - \vec{b}\): - Subtract the components individually: \((v_{ax}, v_{ay}) - (v_{bx}, v_{by}) = (v_{ax} - v_{bx}, v_{ay} - v_{by})\). For vectors \(\vec{a} = (4.0 \mathrm{~m}) \hat{\mathrm{i}} - (3.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b} = (6.0 \mathrm{~m}) \hat{\mathrm{i}} + (8.0 \mathrm{~m}) \hat{\mathrm{j}}\): - The subtraction \(\vec{a} - \vec{b} = (4.0 - 6.0) \hat{\mathrm{i}} + (-3.0 - 8.0) \hat{\mathrm{j}} = -2.0 \hat{\mathrm{i}} - 11.0 \hat{\mathrm{j}}\). This vector represents the difference in position or velocity when moving from \(\vec{b}\) to \(\vec{a}\). Subtraction of vectors is a basic yet powerful operation to find change of state or movement.

In vector mathematics, the resultant vector is the single vector obtained from adding two or more vectors. This principle is heavily used in physics to determine net force, velocity, or displacement. When adding two vectors \(\vec{a}\) and \(\vec{b}\), the resultant vector \(\vec{r}\) is given by component-wise addition: - \(\vec{r} = (a_x + b_x)\hat{\mathrm{i}} + (a_y + b_y)\hat{\mathrm{j}}\). For vectors \(\vec{a} = (4.0 \mathrm{~m}) \hat{\mathrm{i}} - (3.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b} = (6.0 \mathrm{~m}) \hat{\mathrm{i}} + (8.0 \mathrm{~m}) \hat{\mathrm{j}}\), we find the resultant: - \(\vec{r} = (4.0 + 6.0) \hat{\mathrm{i}} + (-3.0 + 8.0) \hat{\mathrm{j}} = 10.0 \hat{\mathrm{i}} + 5.0 \hat{\mathrm{j}}\). The magnitude of the resultant vector can be found using the magnitude formula: \[|\vec{r}| = \sqrt{10.0^2 + 5.0^2} = 11.18 \mathrm{~m}\]. By finding the resultant vector, we can easily determine the total effect of combined forces, displacements, or velocities, making it invaluable in many areas of applied science and engineering.