- Identify the horizontal and vertical components of the given vectors.
- Add the corresponding components to find the result: \((-6 + 2)\vec{i} + (-2 + 4)\vec{j}\).
- Obtain the resulting vector \(-4\vec{i} + 2\vec{j}\).
- The correct choice is \(-4\vec{i} + 2\vec{j}\).
- \(\boxed{-4\vec{i} + 2\vec{j}}\)
Explanation
- Analysis of Vector Summation
To find the sum of two vectors, we combine their corresponding horizontal and vertical components. Given the vectors \(\vec{u} = -6\vec{i} - 2\vec{j}\) and \(\vec{v} = 2\vec{i} + 4\vec{j}\), we express their sum as \(\vec{u} + \vec{v} = (u_x + v_x)\vec{i} + (u_y + v_y)\vec{j}\). By identifying the components \(u_x = -6\), \(u_y = -2\), \(v_x = 2\), and \(v_y = 4\), we can determine the components of the resultant vector.
- Calculation of the Resultant Vector
We compute the sum of the horizontal components \(\vec{i}\): \(-6 + 2 = -4\). Next, we compute the sum of the vertical components \(\vec{j}\): \(-2 + 4 = 2\). Combining these results, the sum of the two vectors is \(-4\vec{i} + 2\vec{j}\). Comparing this to the given options, we find that this matches the second option.
Examples
Vector addition is essential in physics and engineering for calculating the resultant force acting on an object. For example, if two forces, represented as vectors, are pushing a block in different directions, the sum of these vectors tells you the net force and the direction in which the block will accelerate. This principle allows engineers to design stable structures by ensuring that the net force on every component is balanced according to specific safety standards.