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
Here are the steps to solve the problem: Given vectors are m = j + 2k and n = 3i + j. First, write the vectors in component form: m = 0 \\ 1 \\ 2 and n = 3 \\ 1 \\ 0 a) The angle between m and n Step 1: Calculate the dot product of m and n. m · n = (0)(3) + (1)(1) + (2)(0) = 0 + 1 + 0 = 1 Step 2: Calculate the magnitudes of m and n. |m| = sqrt(0^2 + 1^2 + 2^2) = sqrt(0 + 1 + 4) = sqrt(5) |n| = sqrt(3^2 + 1^2 + 0^2) = sqrt(9 + 1 + 0) = sqrt(10) Step 3: Use the dot product formula for the angle between two vectors: = m · n|m| |n|. = (1)/(sqrt(5) · 10) = (1)/(sqrt(50)) = (1)/(5sqrt(2)) To rationalize the denominator: = (1)/(5sqrt(2)) × sqrt(2)sqrt(2) = sqrt(2)10 = (sqrt(2)10) ≈ (0.14142) ≈ 81.87^ The angle between m and n is (sqrt(2)10) ≈ 81.87^. b) The projection of m on n Step 4: Use the formula for the vector projection of m on n: proj_n m = m · n|n|^2 n. We already have m · n = 1 and |n| = sqrt(10), so |n|^2 = 10. Step 5: Substitute the values into the projection formula. proj_n m = (1)/(10) (3i + j) proj_n m = (3)/(10)i + (1)/(10)j The projection of m on n is (3)/(10)i + (1)/(10)j. Send me the next one 📸