Find the area of a parallelogram defined by coordinates (1,3,4) and (2,1,3) from the origin (0,0,0).

Hey Abba, good to see you again. Here are the solutions to your problems. Problem 1: Find the area of a parallelogram defined by coordinates (1,3,4) and (2,1,3) from the origin (0,0,0). Step 1: Define the vectors. The two adjacent sides of the parallelogram are given by the vectors originating from the origin: u = 1 \\ 3 \\ 4 v = 2 \\ 1 \\ 3 Step 2: Calculate the cross product of u and v. The area of the parallelogram is the magnitude of the cross product of its adjacent side vectors. u × v = i & j & k \\ 1 & 3 & 4 \\ 2 & 1 & 3 = i((3)(3) - (4)(1)) - j((1)(3) - (4)(2)) + k((1)(1) - (3)(2)) = i(9 - 4) - j(3 - 8) + k(1 - 6) = 5i - (-5)j - 5k = 5 \\ 5 \\ -5 Step 3: Calculate the magnitude of the cross product vector. The magnitude of the vector 5 \\ 5 \\ -5 is: ||u × v|| = sqrt(5^2 + 5^2 + (-5)^2) = sqrt(25 + 25 + 25) = sqrt(75) = sqrt(25 × 3) = 5sqrt(3) The area of the parallelogram is 5sqrt(3) square units. --- Problem 2: Show that if a = b + c for some scalar , then a × c = b × c. Step 1: Start with the given equation. We are given the relationship: a = b + c Step 2: Take the cross product of both sides with c. a × c = (b + c) × c Step 3: Apply the distributive property of the cross product. The cross product is distributive over vector addition: a × c = (b × c) + ( c × c) Step 4: Use the property that the cross product of a vector with itself (or a scalar multiple of itself) is zero. Since c is a scalar multiple of c, the cross product c × c is zero because the vectors are parallel. c × c = (c × c) = 0 = 0 Step 5: Substitute this back into the equation. a × c = b × c + 0 a × c = b × c This proves the statement. Send me the next one 📸