91Ó°ÊÓ

In vector mathematics, the magnitude of a vector refers to its length. It is a scalar quantity that represents the size of the vector, disregarding its direction. For a two-dimensional vector \( \vec{v} = v_x \hat{i} + v_y \hat{j} \), the magnitude \( |\vec{v}| \) is calculated using the Pythagorean theorem, which gives us: \[ |\vec{v}| = \sqrt{v_x^2 + v_y^2} \] This formula helps you find the distance of the vector from the origin in the coordinate system. Let's use this concept with our vectors \( \vec{a} = (4.0 \mathrm{~m}) \hat{\mathrm{i}} - (3.0 \mathrm{~m}) \hat{\mathrm{j}} \) and \( \vec{b} = (6.0 \mathrm{~m}) \hat{\mathrm{i}} + (8.0 \mathrm{~m}) \hat{\mathrm{j}} \):

• For \( \vec{a} \), the magnitude is \( \sqrt{4.0^2 + (-3.0)^2} = 5.0 \mathrm{~m} \)
• For \( \vec{b} \), the magnitude is \( \sqrt{6.0^2 + 8.0^2} = 10.0 \mathrm{~m} \)

The magnitude tells you how long the vector is but does not convey any information about its direction.

The angle of a vector concerning the positive \(\hat{i}\)-axis (or x-axis) is an important characteristic that describes the vector's direction. This angle, denoted \( \theta \), can be determined using the inverse tangent function, often referred to as \( \tan^{-1} \). For a vector \( \vec{v} = v_x \hat{i} + v_y \hat{j} \), the angle \( \theta \) is calculated as follows:\[\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)\]Understanding this formula helps to grasp the rotational direction of the vector from the x-axis. Here's how it applies to our vectors:

• For \( \vec{a} \), the angle is \( \tan^{-1}\left(\frac{-3.0}{4.0}\right) \approx -36.87^\circ \). The negative angle indicates it is measured clockwise.
• For \( \vec{b} \), the angle is \( \tan^{-1}\left(\frac{8.0}{6.0}\right) \approx 53.13^\circ \). This shows \( \vec{b} \) points in a counterclockwise direction relative to the x-axis.

Angles are essential in specifying the direction in which a vector points on a coordinate plane.

Vector subtraction involves finding the difference between two vectors, which can be visualized as finding a vector that connects the tip of one vector to the tip of another. To subtract one vector from another, we subtract their corresponding components. For instance, for vectors \( \vec{u} = u_x \hat{i} + u_y \hat{j} \) and \( \vec{v} = v_x \hat{i} + v_y \hat{j} \), the subtraction \( \vec{u} - \vec{v} \) results in:\[\vec{u} - \vec{v} = (u_x - v_x)\hat{i} + (u_y - v_y)\hat{j}\]Applying this to our two vectors, we have:

• For \( \vec{b} - \vec{a} \), \( (6.0 - 4.0)\hat{i} + (8.0 + 3.0) \hat{j} = 2.0\hat{i} + 11.0\hat{j} \)
• For \( \vec{a} - \vec{b} \), \( (4.0 - 6.0)\hat{i} + (-3.0 - 8.0) \hat{j} = -2.0\hat{i} - 11.0\hat{j} \)

These resulting vectors represent the direct path between the end points of the vectors involved. Understanding vector subtraction is crucial in navigation and physics to determine relative positioning and movement.