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Let veca = 2hati - hatj + 3hatk , vecb = 3hati - 5hatj + hatk and vecc be a vector such that veca times vecc = vecc times vecb and left(vecmathbfa + vecmathbfcright) . left(vecmathbfb + vecmathbfcright) = 168. Then the maximum value of |vecmathbfc|^2 is:

Solution & Explanation

### Related Formula vecu times vecv = -vecv times vecu textIf vecu times vecv = 0 implies vecu parallel vecv implies vecu = lambda vecv ### Core Logic Given veca times vecc = vecc times vecb implies veca times vecc + vecb times vecc = 0 (veca + vecb) times vecc = 0 implies vecc = lambda(veca + vecb) ### Step 1: Compute a + b veca + vecb = (2+3)hati + (-1-5)hatj + (3+1)hatk = 5hati - 6hatj + 4hatk vecc = lambda(5hati - 6hatj + 4hatk) |vecc|^2 = lambda^2(25 + 36 + 16) = 77lambda^2 ### Step 2: Expand the Dot Product Condition (veca + vecc) cdot (vecb + vecc) = 168 veca cdot vecb + vecc cdot (veca + vecb) + |vecc|^2 = 168 Evaluate veca cdot vecb = (2)(3) + (-1)(-5) + (3)(1) = 6 + 5 + 3 = 14. Substitute vecc cdot (veca + vecb) = lambda |veca + vecb|^2 = 77lambda: 14 + 77lambda + 77lambda^2 = 168 implies 77lambda^2 + 77lambda - 154 = 0 lambda^2 + lambda - 2 = 0 implies lambda = 1 text or lambda = -2 ### Step 3: Maximize |vecc|^2 Maximum value occurs when lambda = -2: |vecc|^2 = 77(-2)^2 = 77 times 4 = 308 ### Pattern Recognition Recognize the cross-product rule inversion immediately: vecx times vecy = vecy times vecz implies (vecx+vecz) parallel vecy. This linear reduction circumvents solving complex linear systems. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Vector Algebra

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