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 are the solutions to your vector problems: Given the position vectors: p = -i + 3j q = 2i + 2j a) Calculate 3p - 4q Step 1: Perform scalar multiplication for each vector. 3p = 3(-i + 3j) = -3i + 9j 4q = 4(2i + 2j) = 8i + 8j Step 2: Subtract the resulting vectors. 3p - 4q = (-3i + 9j) - (8i + 8j) 3p - 4q = (-3 - 8)i + (9 - 8)j 3p - 4q = -11i + j The result is -11i + j. b) Calculate 12p · q Step 1: Calculate the dot product of p and q. p · q = (-1)(2) + (3)(2) p · q = -2 + 6 p · q = 4 Step 2: Multiply the dot product by 12. 12p · q = 12 × 4 12p · q = 48 The result is 48. c) Find the angle between p and q Step 1: Calculate the magnitudes of p and q. |p| = sqrt((-1)^2 + (3)^2) = sqrt(1 + 9) = sqrt(10) |q| = sqrt((2)^2 + (2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2) Step 2: Use the dot product formula for the angle . We already found p · q = 4 from part b. = p · q|p| |q| = (4)/(sqrt(10) · 22) = (4)/(2sqrt(20)) = (4)/(2 · 2sqrt(5)) = (4)/(4sqrt(5)) = (1)/(sqrt(5)) Step 3: Calculate . = ((1)/(sqrt(5))) ≈ 63.43^ The angle between p and q is approximately 63.43^. d) Find the projection of p on q Step 1: Use the formula for vector projection: proj_q p = p · q|q|^2 q. We know p · q = 4 and |q|^2 = (sqrt(8))^2 = 8. proj_q p = (4)/(8) (2i + 2j) Step 2: Simplify the expression. proj_q p = (1)/(2) (2i + 2j) proj_q p = i + j The projection of p on q is i + j. Send me the next one 📸