Let's solve the problem step by step: --- ## **Step 1: Find the Coordinates** - \( A = (a, , b) = (3, , 4) \) - \( C = (, , 4) \) --- ## **Step 2: Find the Vector from A to C** \[ \vec{r}_{AC} = C - A = (-3)\mathbf{i} + (-)\mathbf{j} + (4-4)\mathbf{k} = -3\mathbf{i} + \mathbf{j} + \mathbf{k} \] But this points only along x. Let's check the diagram more carefully: - \( A = (3, , 4) \) - \( C = (, , 8) \) (since the z from A to C rises 4 m above A) So, \[ \vec{r}_{AC} = (-3)\mathbf{i} + (-)\mathbf{j} + (8-4)\mathbf{k} = -3\mathbf{i} + \mathbf{j} + 4\mathbf{k} \] --- ## **Step 3: Find the Unit Vector Along \( \vec{F}_2 \) (\( \vec{F}_2 \) acts along \( \vec{r}_{AC} \))** \[ |\vec{r}_{AC}| = \sqrt{(-3)^2 + ^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Unit vector: \[ \mathbf{u}_{AC} = \frac{\vec{r}_{AC}}{|\vec{r}_{AC}|} = \frac{-3}{5}\mathbf{i} + \mathbf{j} + \frac{4}{5}\mathbf{k} \] --- ## **Step 4: Express \( \vec{F}_2 \) as a Cartesian Vector** \[ \vec{F}_2 = |\vec{F}_2| \cdot \mathbf{u}_{AC} = 700 \left( \frac{-3}{5}\mathbf{i} + \mathbf{j} + \frac{4}{5}\mathbf{k} \right) \] \[ \vec{F}_2 = (-420\mathbf{i} + \mathbf{j} + 560\mathbf{k})\, \text{N} \] --- ## **Step 5: Find the x-component of \( \vec{F}_2 \)** x-component is \(-420\, \text{N}\). --- ## **Final Answers** ### **1. As a Cartesian Vector:** \[ \boxed{ \vec{F}_2 = -420\,\mathbf{i} + \,\mathbf{j} + 560\,\mathbf{k} \text{ N} } \] ### **2. x-component:** \[ \boxed{ (\vec{F}_2)_x = -420\, \text{N} } \] --- **Explanation:** - We found the direction vector from \(A\) to \(C\), normalized it, then multiplied by the magnitude of the force to get the Cartesian vector. - The x-component is the coefficient of \(\mathbf{i}\).

# Explanation of Vector Components ## **Understanding Vectors** Vectors are quantities that have both magnitude and direction. They are represented in a coordinate system using components along the axes (usually x, y, and z). ### **Cartesian Coordinates** In a 3D Cartesian coordinate system, any point or vector can be expressed as: \[ \vec{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k} \] where: - \(V_x\) is the component along the x-axis, - \(V_y\) is the component along the y-axis, - \(V_z\) is the component along the z-axis. ## **Finding Components of a Force** When dealing with forces, we often need to express them in Cartesian form to analyze their effects along different axes. ### **Steps to Determine Components:** 1. **Identify Points:** Determine the coordinates of the points involved (e.g., start point \(A\) and end point \(C\)). 2. **Calculate the Direction Vector:** Subtract the coordinates of the initial point from the final point to find the direction vector. 3. **Normalize the Vector:** Convert the direction vector into a unit vector, which indicates direction without magnitude. 4. **Multiply by Magnitude:** Multiply the unit vector by the force's magnitude to obtain the Cartesian vector representation of the force. 5. **Extract Components:** Read off the components along each axis for further analysis. ## **Example Analysis** In the given problem: - The force \(F_2\) acts along the vector from point \(A\) to point \(C\). - The calculated Cartesian vector was: \[ \vec{F}_2 = -420 \mathbf{i} + 1 \mathbf{j} + 560 \mathbf{k} \text{ N} \] - The x-component of this force is \(-420\, \text{N}\), indicating that it acts in the negative x-direction. ## **Practical Applications** Understanding vector components is crucial in fields like physics, engineering, and computer graphics, where forces and movements must be analyzed in multiple dimensions. ### **Usage Examples:** - **Physics:** Analyzing forces on structures. - **Engineering:** Designing systems that respond to multiple forces. - **Graphics:** Simulating movement in 3D environments. By breaking down forces into their components, we can better understand and predict their behavior in a system.