Vector Algebra

Step 1: Understand the given conditions. We are given that and are unit vectors, which means and . We are also given two vectors and . The condition is that and are perpendicular. Step 2: Use the condition for perpendicular vectors. Two vectors are perpendicular if their dot product is zero. Step 3: Substitute the expressions for and into the dot product equation. Step 4: Expand the dot product using distributive property. Step 5: Use the properties of the dot product: and . Step 6: Substitute the magnitudes of the unit vectors and . Step 7: Use the definition of the dot product to find the angle between and . Substitute the known values: For , the angle is: Step 8: Compare the result with the given options. The angle between and is , which matches option (B).

Step 1: Write down the given vectors and conditions. Conditions: 1. 2. Step 2: Use the dot product condition . Step 3: Use the cross product condition . Calculate : We are given . Equating the components: (Note that (2) and (3) imply (4): ). Step 4: Solve the system of linear equations (1), (2), and (3). From (2), . Substitute this into (3): Now substitute and into equation (1): Step 5: Find x and z using the value of y. Step 6: State the vector b and compare with options. This result matches option (D).

We are given the vectors: - - The angle between two vectors and can be found using the formula for the dot product: First, calculate the dot product . Since and , we have: Next, calculate the magnitudes of and : Now, substitute these values into the formula for : Thus, , which means: Therefore, the angle between and is .