1. **Homework 1: Determine if two lines are perpendicular, skew, or intersecting** - Problem: Given two lines in 3D or 2D space, determine if they are perpendicular, skew, or intersecting. - Formula: Two lines are perpendicular if their direction vectors satisfy $\mathbf{d_1} \cdot \mathbf{d_2} = 0$. - Skew lines do not intersect and are not parallel; intersecting lines meet at a point. Steps: 1. Identify direction vectors $\mathbf{d_1}$ and $\mathbf{d_2}$ of the two lines. 2. Compute the dot product $\mathbf{d_1} \cdot \mathbf{d_2}$. 3. If dot product is zero, lines are perpendicular. 4. Check if lines intersect by solving their parametric equations. 5. If no intersection and not parallel, lines are skew. 2. **Homework 2: Determine the position of a line** - Problem: Find the position of a line relative to coordinate axes or other lines. - Important: Position can be parallel, intersecting, or skew. Steps: 1. Write the line equation in parametric or vector form. 2. Compare with axes or other lines to check parallelism or intersection. 3. Use direction vectors and points to analyze relative position. 3. **Homework 3: Find distance (projection and true length) from point K to line AB** - Problem: Calculate the shortest distance from point $K$ to line $AB$ and the projection length. - Formula: Distance $d = \frac{|(\mathbf{K} - \mathbf{A}) \times \mathbf{AB}|}{|\mathbf{AB}|}$ - Projection length $p = \frac{(\mathbf{K} - \mathbf{A}) \cdot \mathbf{AB}}{|\mathbf{AB}|}$ Steps: 1. Find vectors $\mathbf{K} - \mathbf{A}$ and $\mathbf{AB}$. 2. Compute cross product magnitude $|(\mathbf{K} - \mathbf{A}) \times \mathbf{AB}|$. 3. Calculate distance $d$ using the formula. 4. Compute dot product $(\mathbf{K} - \mathbf{A}) \cdot \mathbf{AB}$. 5. Calculate projection length $p$. These steps help analyze line relationships and distances in coordinate geometry.