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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Here's the solution for question 9: To determine if vectors are orthogonal, we check if their dot product is zero. If they are orthogonal, we then check if they are orthonormal by verifying if both vectors are unit vectors (i.e., their magnitudes are 1). i) Given vectors a = (4, 2, -1) and b = (1, -3, -2). Step 1: Calculate the dot product a · b. a · b = (4)(1) + (2)(-3) + (-1)(-2) a · b = 4 - 6 + 2 a · b = 0 Step 2: Determine if the vectors are orthogonal. Since a · b = 0, the vectors a and b are orthogonal. Step 3: Calculate the magnitudes of a and b. ||a|| = sqrt(4^2 + 2^2 + (-1)^2) = sqrt(16 + 4 + 1) = sqrt(21) ||b|| = sqrt(1^2 + (-3)^2 + (-2)^2) = sqrt(1 + 9 + 4) = sqrt(14) Step 4: Determine if the vectors are orthonormal. Since ||a|| = sqrt(21) ≠ 1 and ||b|| = sqrt(14) ≠ 1, the vectors are not unit vectors. Therefore, the vectors a and b are not orthonormal. ii) Given vectors a = (5, -1, 3) and b = (4, 7, -2). Step 1: Calculate the dot product a · b. a · b = (5)(4) + (-1)(7) + (3)(-2) a · b = 20 - 7 - 6 a · b = 13 - 6 a · b = 7 Step 2: Determine if the vectors are orthogonal. Since a · b = 7 ≠ 0, the vectors a and b are not orthogonal. Step 3: Determine if the vectors are orthonormal. Since the vectors are not orthogonal, they cannot be orthonormal. Final Answer: i) The vectors a = (4, 2, -1) and b = (1, -3, -2) are orthogonal but not orthonormal. ii) The vectors a = (5, -1, 3) and b = (4, 7, -2) are not orthogonal and therefore not orthonormal. Send me the next one 📸