# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - **Vector addition and subtraction** - **Component form of vectors** - **Collinearity of points** - **Parallel vectors** - **Dot product and orthogonality** - **Position vectors and their properties** Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: - **Vector addition and subtraction**: $$ \vec{A} + \vec{B} = (x_1 + x_2, y_1 + y_2) $$ $$ \vec{A} - \vec{B} = (x_1 - x_2, y_1 - y_2) $$ - **Dot product**: $$ \vec{A} \cdot \vec{B} = x_1 \cdot x_2 + y_1 \cdot y_2 $$ - **Magnitude of a vector**: $$ |\vec{A}| = \sqrt{x_1^2 + y_1^2} $$ - **Collinearity check using vectors**: If vectors $\vec{A}$ and $\vec{B}$ are collinear, then: $$ \vec{A} = k \cdot \vec{B} \quad \text{for some scalar} \, k. $$ --- # **Summary** This exercise covers a range of vector operations, including addition, subtraction, and finding components. Additionally, the concepts of parallelism and collinearity are explored. The key challenge lies in applying vector operations to solve geometrical problems, such as proving that points are collinear or determining the relationship between vectors in geometric configurations like parallelograms and triangles. --- # **Reference Videos** ![](https://www.youtube.com/watch?v=yLVEdMbTJdc&pp=ygUTbmJmIGV4IDMuMSBjbGFzcyAxMQ%3D%3D) ![](https://www.youtube.com/watch?v=yxX1ne4o68Y&pp=ygUTbmJmIGV4IDMuMSBjbGFzcyAxMQ%3D%3D) ---