Practical Examples
5. Applying Our Knowledge with Real-World Scenarios
Okay, enough theory! Let’s put these techniques into action with a few practical examples. This is where the rubber meets the road, and you’ll see exactly how to apply these concepts to real-world scenarios.
Example 1: 2D Vectors Imagine we have vector a = (3, 6) and vector b = (1, 2). Let’s use the scalar multiple method. We want to find a k such that (3, 6) = k(1, 2). This gives us two equations: 3 = k(1) and 6 = k(2). Solving each, we get k = 3 in both cases. Since the value of k is consistent, vectors a and b are parallel. Using the determinant method, we get (3 2) – (1 6) = 6 – 6 = 0, confirming that they are parallel too!
Example 2: 3D Vectors Let’s say we have a = (2, 4, 6) and b = (1, 2, 3). To prove theyre parallel, we calculate their cross product. a x b = ((4 3) – (6 2), (6 1) – (2 3), (2 2) – (4 1)) = (12 – 12, 6 – 6, 4 – 4) = (0, 0, 0). Since the cross product is the zero vector, vectors a and b are parallel.
Example 3: Are these vectors parallel? Let’s examine a = (1, 5) and b = (2, 7). From the scalar multiple method, 1 = k2 and 5 = k7. From the first equation, k = 1/2. From the second, k = 5/7. As 1/2 is not equal to 5/7, these vectors are not parallel.
These examples demonstrate how to quickly and accurately determine if two vectors are parallel, regardless of their dimensionality. Understanding the underlying principles and applying the appropriate method is key to solving these types of problems efficiently and correctly.