If vector a = 2i - j - k and vector b = λi + j + 5k and a ⊥ b, then λ = ?

Here are the solutions for questions 16, 17, and 18. Question 16: If a = 2i - j - k and b = + j + 5k and a b, then = \_ If two vectors are perpendicular, their dot product is zero (a · b = 0). Step 1: Calculate the dot product of a and b. a · b = (2)() + (-1)(1) + (-1)(5) a · b = 2 - 1 - 5 a · b = 2 - 6 Step 2: Set the dot product to zero and solve for . 2 - 6 = 0 2 = 6 = (6)/(2) = 3 The final answer is (c). Question 17: Vector projection a = i on b = j is The formula for the vector projection of a on b is proj_b a = ( a · b|b|^2 ) b. Given: a = i = 1i + 0j + 0k b = j = 0i + 1j + 0k Step 1: Calculate the dot product a · b. a · b = (1)(0) + (0)(1) + (0)(0) a · b = 0 Step 2: Calculate the magnitude squared of b, |b|^2. |b|^2 = (0)^2 + (1)^2 + (0)^2 = 1 Step 3: Calculate the vector projection. proj_b a = ( (0)/(1) ) j proj_b a = 0 · j proj_b a = 0 The final answer is (b). Question 18: Scalar projection vector i - j - k on vector 3i + j + 3k is The formula for the scalar projection of u on v is comp_v u = u · v|v|. Given: u = i - j - k v = 3i + j + 3k Step 1: Calculate the dot product u · v. u · v = (1)(3) + (-1)(1) + (-1)(3) u · v = 3 - 1 - 3 u · v = -1 Step 2: Calculate the magnitude of v, |v|. |v| = sqrt((3)^2 + (1)^2 + (3)^2) |v| = sqrt(9 + 1 + 9) |v| = sqrt(19) Step 3: Calculate the scalar projection. comp_v u = (-1)/(sqrt(19)) Comparing this result with the given options (a) 0, (b) 0, (c) k, (d) None, our calculated scalar value (-1)/(sqrt(19)) does not match any of the options (a), (b), or (c). The final answer is (d). 3 done, 2 left today. You're making progress.