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
To find the unit vector perpendicular to two vectors, we first calculate their cross product, which gives a vector perpendicular to both. Then, we divide this resultant vector by its magnitude to get the unit vector. From the image, the two vectors appear to be: a = i + 2j + k b = 5i - j + 2k Step 1: Calculate the cross product a × b. a × b = i & j & k \\ 1 & 2 & 1 \\ 5 & -1 & 2 = i((2)(2) - (1)(-1)) - j((1)(2) - (1)(5)) + k((1)(-1) - (2)(5)) = i(4 - (-1)) - j(2 - 5) + k(-1 - 10) = i(5) - j(-3) + k(-11) = 5i + 3j - 11k Step 2: Calculate the magnitude of the cross product |a × b|. Let c = 5i + 3j - 11k. |c| = sqrt((5)^2 + (3)^2 + (-11)^2) = sqrt(25 + 9 + 121) = sqrt(155) Step 3: Compute the unit vector perpendicular to both a and b. The unit vector n is given by a × b|a × b|. n = 5i + 3j - 11ksqrt(155) n = (5)/(sqrt(155))i + (3)/(sqrt(155))j - (11)/(sqrt(155))k The unit vector perpendicular to both vectors is (5)/(sqrt(155))i + (3)/(sqrt(155))j - (11)/(sqrt(155))k. What's next?