The angle that the vector `bar (OA)= 5hati + 5hatj` mix with y-axis is

Options

A. 30°
B. 45°
C. 60°
D. `tan^(-1)(1/2)`

Answer

To find the angle that the vector \(\vec{OA} = 5\hat{i} + 5\hat{j}\) makes with the y-axis, we can follow these steps:

### Step 1: Understand the Vector
The vector \(\vec{OA} = 5\hat{i} + 5\hat{j}\) indicates that it has components of 5 units in the x-direction (i.e., along the x-axis) and 5 units in the y-direction (i.e., along the y-axis).

### Step 2: Draw the Vector
Visualize the vector in a coordinate system. The x-axis is horizontal, and the y-axis is vertical. The point A (5, 5) can be plotted on this graph. The vector \(\vec{OA}\) starts from the origin (0, 0) and ends at point A (5, 5).

### Step 3: Identify the Right Triangle
The vector forms a right triangle with the x-axis and y-axis. The horizontal leg (base) of the triangle is 5 units (along the x-axis), and the vertical leg (perpendicular) is also 5 units (along the y-axis).

### Step 4: Use Trigonometric Ratios
To find the angle \(\theta\) that the vector makes with the y-axis, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, the opposite side is the x-component (5 units), and the adjacent side is the y-component (5 units).

### Step 5: Calculate the Tangent
Substituting the values into the tangent function:
\[ \tan(\theta) = \frac{5}{5} = 1 \]

### Step 6: Find the Angle
Now, we need to find the angle \(\theta\) such that:
\[ \tan(\theta) = 1 \]
The angle whose tangent is 1 is \(45^\circ\).

### Conclusion
Thus, the angle that the vector \(\vec{OA}\) makes with the y-axis is \(45^\circ\).

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