Let `veca, vecb and vecc` are three non - collinear vectors in a plane such that `|veca|=2, |vecb|=5 and |vecc|=sqrt(29)`. If the angle between `veca and vecc` is `theta_(1)` and the angle between `vecb and vecc` is `theta_(2)`, where `theta_(1), theta_(2) in [(pi)/(2),pi]`, then the value of `theta_(1)+theta_(2)` is equal to

Options

A. `(7pi)/(6)`
B. `(4pi)/(6)`
C. `(3pi)/(2)`
D. `(7pi)/(4)`

Answer

To solve the problem step-by-step, we will analyze the given vectors and their magnitudes, and then apply the properties of triangles and angles.

### Step 1: Write down the given information
We have three non-collinear vectors:
- \( |\vec{a}| = 2 \)
- \( |\vec{b}| = 5 \)
- \( |\vec{c}| = \sqrt{29} \)

### Step 2: Apply the Pythagorean theorem
Since the vectors are non-collinear and their magnitudes satisfy the equation:
\[ |\vec{a}|^2 + |\vec{b}|^2 = |\vec{c}|^2 \]
we can check:
\[ 2^2 + 5^2 = \sqrt{29}^2 \]
Calculating the left side:
\[ 4 + 25 = 29 \]
Calculating the right side:
\[ 29 = 29 \]
This confirms that the vectors form a right triangle.

### Step 3: Understand the angles
In a right triangle, the angles opposite the sides can be defined. Here, we denote:
- \( \theta_1 \) as the angle between \( \vec{a} \) and \( \vec{c} \)
- \( \theta_2 \) as the angle between \( \vec{b} \) and \( \vec{c} \)

Since \( \vec{c} \) is the hypotenuse, the angles \( \theta_1 \) and \( \theta_2 \) are complementary to the right angle.

### Step 4: Relate the angles
From the properties of triangles, we know:
\[ \theta_1 + \theta_2 = \frac{\pi}{2} \]
However, since the problem states that \( \theta_1, \theta_2 \in \left[\frac{\pi}{2}, \pi\right] \), we need to adjust our understanding.

### Step 5: Find the sum of angles
Since \( \theta_1 \) and \( \theta_2 \) are the angles in the triangle formed with the right angle, the total angle around point \( C \) (the vertex opposite the hypotenuse) can be expressed as:
\[ \theta_1 + \theta_2 + \frac{\pi}{2} = \pi \]
Thus, rearranging gives:
\[ \theta_1 + \theta_2 = \pi - \frac{\pi}{2} \]
This simplifies to:
\[ \theta_1 + \theta_2 = \frac{3\pi}{2} \]

### Conclusion
The value of \( \theta_1 + \theta_2 \) is:
\[ \theta_1 + \theta_2 = \frac{3\pi}{2} \]