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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1a) Step 1: Find the magnitude of vector a. Given a = 3i - 4j + 5k, its components are (3, -4, 5). The magnitude |a| is given by: |a| = sqrt(3^2 + (-4)^2 + 5^2) |a| = sqrt(9 + 16 + 25) |a| = sqrt(50) |a| = 5sqrt(2) Step 2: Find the unit vector in the direction of a. The unit vector a is given by (a)/(|a|). a = (1)/(5sqrt(2))(3i - 4j + 5k) a = (3)/(5sqrt(2))i - (4)/(5sqrt(2))j + (5)/(5sqrt(2))k Rationalize the denominators: a = 3sqrt(2)10i - 4sqrt(2)10j + 5sqrt(2)10k a = 3sqrt(2)10i - 2sqrt(2)5j + sqrt(2)2k The unit vector is 3sqrt(2)10i - 2sqrt(2)5j + sqrt(2)2k. 1b) Step 1: Compute the dot product a · b. Given a = 2i - j + 3k and b = i + 4j - 2k. a · b = (2)(1) + (-1)(4) + (3)(-2) a · b = 2 - 4 - 6 a · b = -8 Step 2: Compute the magnitudes of a and b. |a| = sqrt(2^2 + (-1)^2 + 3^2) = sqrt(4 + 1 + 9) = sqrt(14) |b| = sqrt(1^2 + 4^2 + (-2)^2) = sqrt(1 + 16 + 4) = sqrt(21) Step 3: Compute the angle between a and b. Using the formula = (a · b)/(|a||b|): = (-8)/(sqrt(14)21) = (-8)/(sqrt(294)) = (-8)/(7sqrt(6)) Rationalize the denominator: = -8sqrt(6)42 = -4sqrt(6)21 = (-4sqrt(6)21) ≈ (-0.4666) ≈ 117.8^ The dot product is -8 and the angle between them is approximately 117.8^. 1c) Step 1: Identify vectors a and b. Given a = i + 2j - k and b = 3i - j + 2k. Step 2: Compute the dot product a · b. a · b = (1)(3) + (2)(-1) + (-1)(2) a · b = 3 - 2 - 2 a · b = -1 Step 3: Compute the magnitude squared of b, |b|^2. |b|^2 = 3^2 + (-1)^2 + 2^2 |b|^2 = 9 + 1 + 4 |b|^2 = 14 Step 4: Compute the vector projection of a onto b. The formula for the vector projection of a onto b is proj_b a = (a · b)/(|b|^2) b. proj_b a = (-1)/(14)(3i - j + 2k) proj_b a = -(3)/(14)i + (1)/(14)j - (2)/(14)k proj_b a = -(3)/(14)i + (1)/(14)j - (1)/(7)k The vector projection is -(3)/(14)i + (1)/(14)j - (1)/(7)k. 2a) Step 1: Identify vectors u and v. Given u = (1, -2, 3) and v = (4, 0, -1). Step 2: Compute the cross product u × v. u × v = i & j & k \\ 1 & -2 & 3 \\ 4 & 0 & -1 u × v = i((-2)(-1) - (3)(0)) - j((1)(-1) - (3)(4)) + k((1)(0) - (-2)(4)) u × v = i(2 - 0) - j(-1 - 12) + k(0 - (-8)) u × v = 2i - j(-13) + k(8) u × v = 2i + 13j + 8k The cross product is 2i + 13j + 8k. 2b) Step 1: Identify vectors p and q. Given p = 2i + j - k and q = i - 3j + 4k. Step 2: Compute the cross product p × q. p × q = i & j & k \\ 2 & 1 & -1 \\ 1 & -3 & 4 p × q = i((1)(4) - (-1)(-3)) - j((2)(4) - (-1)(1)) + k((2)(-3) - (1)(1)) p × q = i(4 - 3) - j(8 - (-1)) + k(-6 - 1) p × q = i(1) - j(9) + k(-7) p × q = i - 9j - 7k Step 3: Find the magnitude of the cross product. The area of the parallelogram is |p × q|. |p × q| = sqrt(1^2 + (-9)^2 + (-7)^2) |p × q| = sqrt(1 + 81 + 49) |p × q| = sqrt(131) The area of the parallelogram is sqrt(131) square units. 2c) Step 1: Form a matrix with the given vectors as rows. Given a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9). Let M = 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 . Step 2: Calculate the determinant of the matrix M. If (M) = 0, the vectors are linearly dependent. If (M) ≠ 0, they are linearly independent. (M) = 1(5 · 9 - 6 · 8) - 2(4 · 9 - 6 · 7) + 3(4 · 8 - 5 · 7) (M) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) (M) = 1(-3) - 2(-6) + 3(-3) (M) = -3 + 12 - 9 (M) = 0 Since the determinant is 0, the vectors are linearly dependent. The vectors are linearly dependent. 3a) Step 1: Compute AB. A = 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 , B = 2 & 0 & 1 \\ -1 & 1 & 0 \\ 3 & 2 & 1 AB = (1)(2)+(2)(-1)+(3)(3) & (1)(0)+(2)(1)+(3)(2) & (1)(1)+(2)(0)+(3)(1) \\ (0)(2)+(1)(-1)+(4)(3) & (0)(0)+(1)(1)+(4)(2) & (0)(1)+(1)(0)+(4)(1) \\ (5)(2)+(6)(-1)+(0)(3) & (5)(0)+(6)(1)+(0)(2) & (5)(1)+(6)(0)+(0)(1) AB = 2-2+9 & 0+2+6 & 1+0+3 \\ 0-1+12 & 0+1+8 & 0+0+4 \\ 10-6+0 & 0+6+0 & 5+0+0 AB = 9 & 8 & 4 \\ 11 & 9 & 4 \\ 4 & 6 & 5 Step 2: Compute BA. BA = 2 & 0 & 1 \\ -1 & 1 & 0 \\ 3 & 2 & 1 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 BA = (2)(1)+(0)(0)+(1)(5) & (2)(2)+(0)(1)+(1)(6) & (2)(3)+(0)(4)+(1)(0) \\ (-1)(1)+(1)(0)+(0)(5) & (-1)(2)+(1)(1)+(0)(6) & (-1)(3)+(1)(4)+(0)(0) \\ (3)(1)+(2)(0)+(1)(5) & (3)(2)+(2)(1)+(1)(6) & (3)(3)+(2)(4)+(1)(0) BA = 2+0+5 & 4+0+6 & 6+0+0 \\ -1+0+0 & -2+1+0 & -3+4+0 \\ 3+0+5 & 6+2+6 & 9+8+0 BA = 7 & 10 & 6 \\ -1 & -1 & 1 \\ 8 & 14 & 17 Step 3: Compute AB - BA. AB - BA = 9 & 8 & 4 \\ 11 & 9 & 4 \\ 4 & 6 & 5 - 7 & 10 & 6 \\ -1 & -1 & 1 \\ 8 & 14 & 17 AB - BA = 9-7 & 8-10 & 4-6 \\ 11-(-1) & 9-(-1) & 4-1 \\ 4-8 & 6-14 & 5-17 AB - BA = 2 & -2 & -2 \\ 12 & 10 & 3 \\ -4 & -8 & -12 The result is 2 & -2 & -2 \\ 12 & 10 & 3 \\ -4 & -8 & -12 . 3b) Step 1: Identify matrix C. C = 2 & 1 & 3 \\ 4 & 0 & 2 \\ 1 & 5 &