Unit Vector in the XY-plane
• To find a unit vector in the XY-plane that makes an angle of 45 degrees with the vector i-j and an angle of 60 degrees with the vector 3i-4j, we can follow these steps.

Step 1: Find the Unit Vector of i-j
• The vector i-j can be represented as (1, -1) in component form.
• To find the unit vector of i-j, we divide the vector by its magnitude.
• The magnitude of i-j is √(1^2 + (-1)^2) = √2.
• Therefore, the unit vector of i-j is (1/√2, -1/√2).

Step 2: Find the Unit Vector of 3i-4j
• The vector 3i-4j can be represented as (3, -4) in component form.
• To find the unit vector of 3i-4j, we divide the vector by its magnitude.
• The magnitude of 3i-4j is √(3^2 + (-4)^2) = 5.
• Therefore, the unit vector of 3i-4j is (3/5, -4/5).

Step 3: Find the Unit Vector in the XY-plane
• Let the unit vector in the XY-plane be represented as (a, b).
• Since the vector makes a 45-degree angle with i-j, we can set up the equation (a/√2, b/√2) • (1/√2, -1/√2) = cos(45).
• Similarly, for the 60-degree angle with 3i-4j, we have (a/√2, b/√2) • (3/5, -4/5) = cos(60).
• Solve these two equations simultaneously to find the values of a and b.
• Finally, normalize the vector (a, b) to get the unit vector in the XY-plane.