This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Given the vectors: a = 2i + 5j - k b = 3i - 2j + 4k We need to find (i) a × b, (ii) b × a, and (iii) (a + b) × (a - b). i) Find a × b. Step 1: Set up the determinant for the cross product. The cross product a × b is calculated as: a × b = i & j & k \\ 2 & 5 & -1 \\ 3 & -2 & 4 Step 2: Calculate the components of the cross product. a × b = ( (5)(4) - (-1)(-2) )i - ( (2)(4) - (-1)(3) )j + ( (2)(-2) - (5)(3) )k a × b = (20 - 2)i - (8 - (-3))j + (-4 - 15)k a × b = 18i - (8 + 3)j - 19k a × b = 18i - 11j - 19k The result for (i) is 18i - 11j - 19k. ii) Find b × a. Step 1: Use the property of cross products that b × a = -(a × b). Using the result from part (i): b × a = -(18i - 11j - 19k) b × a = -18i + 11j + 19k The result for (ii) is -18i + 11j + 19k. iii) Find (a + b) × (a - b). Step 1: Calculate a + b. a + b = (2i + 5j - k) + (3i - 2j + 4k) a + b = (2+3)i + (5-2)j + (-1+4)k a + b = 5i + 3j + 3k Step 2: Calculate a - b. a - b = (2i + 5j - k) - (3i - 2j + 4k) a - b = (2-3)i + (5-(-2))j + (-1-4)k a - b = -1i + 7j - 5k Step 3: Set up the determinant for the cross product of (a + b) and (a - b). (a + b) × (a - b) = i & j & k \\ 5 & 3 & 3 \\ -1 & 7 & -5 Step 4: Calculate the components of the cross product. (a + b) × (a - b) = ( (3)(-5) - (3)(7) )i - ( (5)(-5) - (3)(-1) )j + ( (5)(7) - (3)(-1) )k (a + b) × (a - b) = (-15 - 21)i - (-25 - (-3))j + (35 - (-3))k (a + b) × (a - b) = -36i - (-25 + 3)j + (35 + 3)k (a + b) × (a - b) = -36i - (-22)j + 38k (a + b) × (a - b) = -36i + 22j + 38k The result for (iii) is -36i + 22j + 38k. 3 done, 2 left today. You're making progress.