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 solutions to your questions: 1. Find the angle between the vectors a = 2i + 2j - k and b = 6i - 3j + 2k. To find the angle between two vectors, we use the formula = a · b|a| |b|. Step 1: Calculate the dot product a · b. a · b = (2)(6) + (2)(-3) + (-1)(2) = 12 - 6 - 2 = 4 Step 2: Calculate the magnitude of a. |a| = sqrt((2)^2 + (2)^2 + (-1)^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3 Step 3: Calculate the magnitude of b. |b| = sqrt((6)^2 + (-3)^2 + (2)^2) = sqrt(36 + 9 + 4) = sqrt(49) = 7 Step 4: Calculate and then . = (4)/((3)(7)) = (4)/(21) = ((4)/(21)) The angle between the vectors is ((4)/(21)). 2. Find the projection of the vector a = 2i - 3j + 6k on the vector b = i + 2j + 2k. The scalar projection of vector a onto vector b is given by the formula comp_ba = a · b|b|. Step 1: Calculate the dot product a · b. a · b = (2)(1) + (-3)(2) + (6)(2) = 2 - 6 + 12 = 8 Step 2: Calculate the magnitude of b. |b| = sqrt((1)^2 + (2)^2 + (2)^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3 Step 3: Calculate the scalar projection. comp_ba = (8)/(3) The projection of a on b is (8)/(3). 3. Find the area of the parallelogram whose adjacent sides are given by the vectors a = 3i + j + 4k and b = i - j + k. The area of a parallelogram with adjacent sides a and b is given by the magnitude of their cross product, |a × b|. Step 1: Calculate the cross product a × b. a × b = i & j & k \\ 3 & 1 & 4 \\ 1 & -1 & 1 = i((1)(1) - (4)(-1)) - j((3)(1) - (4)(1)) + k((3)(-1) - (1)(1)) = i(1 - (-4)) - j(3 - 4) + k(-3 - 1) = i(5) - j(-1) + k(-4) = 5i + j - 4k Step 2: Calculate the magnitude of a × b. |a × b| = sqrt((5)^2 + (1)^2 + (-4)^2) = sqrt(25 + 1 + 16) = sqrt(42) The area of the parallelogram is sqrt(42) square units. 4. Find the area of the triangle whose adjacent sides are given by the vectors a = 3i + j + 4k and b = i - j + k. The area of a triangle with adjacent sides a and b is half the magnitude of their cross product, (1)/(2)|a × b|. We already calculated a × b and its magnitude in Question 3. Step 1: Use the magnitude of the cross product from Question 3. |a × b| = sqrt(42) Step 2: Calculate the area of the triangle. Area = (1)/(2) |a × b| Area = (1)/(2) sqrt(42) The area of the triangle is sqrt(42)2 square units. What's next?