To find the angle between the vector i + 3j and the positive direction of the x-axis, we can use the dot product formula. The dot product of two vectors A and B is given by:
A · B = |A| |B| cos(θ)
where θ is the angle between them.
In this case, our vector is A = i + 3j, and we want to find the angle θ with respect to the positive x-axis, represented by the vector B = i.
First, we can express our vectors in component form:
- A = (1, 3)
- B = (1, 0)
Next, we calculate the dot product:
A · B = (1)(1) + (3)(0) = 1
Then, we find the magnitudes of the vectors:
|A| = √(1² + 3²) = √(1 + 9) = √10
|B| = √(1² + 0²) = √1 = 1
Now we can substitute these values into the dot product formula:
1 = √10 * 1 * cos(θ)
This simplifies to:
cos(θ) = 1/√10
To find the angle θ, we take the arccosine:
θ = arccos(1/√10)
Calculating this will give us the angle in radians or degrees, based on the calculator used. Therefore, the angle between the vector i + 3j and the positive x-axis is:
θ ≈ 71.57°