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's the solution for question 4. 4. For each of the following vectors, find: a) The magnitude of the vector b) The unit vector in the direction of the vector i. 6i + 8j a) The magnitude of the vector Step 1: Let the vector be v = 6i + 8j. The magnitude of a vector xi + yj is given by |v| = sqrt(x^2 + y^2). |v| = sqrt((6)^2 + (8)^2) Step 2: Calculate the squares and sum them. |v| = sqrt(36 + 64) = sqrt(100) Step 3: Find the square root. |v| = 10 The magnitude of the vector is 10. b) The unit vector in the direction of the vector Step 1: The unit vector v in the direction of v is given by v = v|v|. v = 6i + 8j10 Step 2: Divide each component by the magnitude. v = (6)/(10)i + (8)/(10)j Step 3: Simplify the fractions. v = (3)/(5)i + (4)/(5)j The unit vector is (3)/(5)i + (4)/(5)j. ii. -3i + 5j a) The magnitude of the vector Step 1: Let the vector be u = -3i + 5j. The magnitude is given by |u| = sqrt(x^2 + y^2). |u| = sqrt((-3)^2 + (5)^2) Step 2: Calculate the squares and sum them. |u| = sqrt(9 + 25) = sqrt(34) The magnitude of the vector is sqrt(34). b) The unit vector in the direction of the vector Step 1: The unit vector u in the direction of u is given by u = u|u|. u = -3i + 5jsqrt(34) Step 2: Divide each component by the magnitude. u = -(3)/(sqrt(34))i + (5)/(sqrt(34))j The unit vector is -(3)/(sqrt(34))i + (5)/(sqrt(34))j. That's 2 down. 3 left today — send the next one.