Introduction:

To find a unit vector in the xy-plane that makes a specific angle with two given vectors, we can use the concept of dot product and trigonometry. The dot product can help us determine the angle between two vectors, and trigonometry can help us find the components of the unit vector.

Given:
- Vector A = i + j (makes an angle of 45° with the unit vector)
- Vector B = 3i - 4j (makes an angle of 60° with the unit vector)

Step 1: Finding the Angle between the Unit Vector and Vector A:

To find the angle between the unit vector and vector A, we can use the dot product formula:

A · B = |A| |B| cos θ

Since the unit vector has a magnitude of 1, the dot product simplifies to:

1 · 1 cos 45° = cos 45°

Therefore, the angle between the unit vector and vector A is 45°.

Step 2: Finding the Angle between the Unit Vector and Vector B:

Similarly, we can find the angle between the unit vector and vector B using the dot product formula:

A · B = |A| |B| cos θ

1 · √(3² + (-4)²) cos 60° = √(3² + (-4)²) cos 60°

Therefore, the angle between the unit vector and vector B is cos 60°.

Step 3: Finding the Components of the Unit Vector:

Let the components of the unit vector be (x, y). Since it lies in the xy-plane, the z-component is 0.

We can now use the angles we found in Step 1 and Step 2 to set up the following equations:

cos 45° = x / √(x² + y²)
cos 60° = (3x - 4y) / √(x² + y²)

Simplifying these equations, we get:

1 / √2 = x / √(x² + y²)
1 / 2 = (3x - 4y) / √(x² + y²)

Squaring both sides of the equations, we have:

1 / 2 = (x² + y²) / (x² + y²)
1 / 4 = (9x² - 24xy + 16y²) / (x² + y²)

Combining the equations, we get:

4x² - 8xy + 4y² = 9x² - 24xy + 16y²

Simplifying further, we have:

5x² + 16xy + 12y² = 0

Step 4: Solving the Quadratic Equation:

To solve the quadratic equation above, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

By comparing the equation with ax² + bx + c = 0, we can determine the values of a, b, and c:

a = 5
b = 16
c = 12

Solving the quadratic equation, we find two sets of solutions for (