To determine the angle between the vector 3i + 3j and the positive direction of the x-axis, we can use the concept of the dot product and the formula that relates the two vectors.
First, let’s express the vector similarly to its components. The vector 3i + 3j can be represented as:
- V = (3, 3)
Now, the positive direction of the x-axis can be represented as the unit vector:
- U = (1, 0)
To find the angle θ between the two vectors, we can use the following formula:
cos(θ) = (V • U) / (|V| * |U|)
Where:
- V • U is the dot product of the vectors.
- |V| is the magnitude of vector V.
- |U| is the magnitude of vector U.
Let’s compute these values:
- The dot product: V • U = 3 * 1 + 3 * 0 = 3.
- Magnitude of vector V: |V| = √(3² + 3²) = √(9 + 9) = √18 = 3√2.
- Magnitude of vector U: |U| = √(1² + 0²) = √1 = 1.
Now, substitute these values into the cosine formula:
cos(θ) = 3 / (3√2 * 1) = 1 / √2
To find the angle θ, we compute the arccosine:
θ = arccos(1 / √2)
This gives us:
- θ = 45°
Therefore, the angle between the vector 3i + 3j and the positive direction of the x-axis is 45 degrees.