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.
ScanSolve AI Answer
Step 1: For two vectors to be perpendicular, their dot product must be zero. Let the given vectors be a and b. a = t \\ 4 \\ 2t+1 , b = t+2 \\ 1-t \\ -1 The dot product a · b is calculated as: t(t+2) + 4(1-t) + (2t+1)(-1) = 0 Step 2: Expand the terms in the equation. t^2 + 2t + 4 - 4t - 2t - 1 = 0 Step 3: Combine like terms to simplify the equation into a quadratic form. t^2 + (2t - 4t - 2t) + (4 - 1) = 0 t^2 - 4t + 3 = 0 Step 4: Solve the quadratic equation for t. We can factor the quadratic equation. We need two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. (t-1)(t-3) = 0 Step 5: Set each factor equal to zero to find the possible values of t. t-1 = 0 t = 1 t-3 = 0 t = 3 The possible values for t are 1 and 3. The possible values of the constant t are 1 and 3. Send me the next one 📸