The resultant of `vec(A)` and `vec(B)` makes an angle `alpha` with `vec(A)` and `omega` with `vec(B)` , then:-

Options

A. `alpha lt beta`
B. `alpha lt beta` if `A lt B`
C. `alpha lt beta` if `A gt B`
D. `alpha lt beta`if `A=B`

Explanation

To solve the problem, we need to analyze the relationship between the angles and the magnitudes of the vectors involved. Let's break it down step by step.

### Step 1: Understand the Vectors and Angles
We have two vectors, \(\vec{A}\) and \(\vec{B}\). The resultant vector \(\vec{R}\) of these two vectors makes an angle \(\alpha\) with \(\vec{A}\) and an angle \(\omega\) with \(\vec{B}\).

### Step 2: Use the Law of Sines
According to the law of sines in a triangle formed by the vectors, we can relate the angles and the sides opposite to them. The law states that:

\[ \frac{R}{\sin(\omega)} = \frac{B}{\sin(\alpha)} \]

This means that the ratio of the length of the resultant vector \(\vec{R}\) to the sine of the angle \(\omega\) is equal to the ratio of the length of vector \(\vec{B}\) to the sine of the angle \(\alpha\).

### Step 3: Apply the Law of Cosines
We can also use the law of cosines to relate the sides and angles in the triangle formed by \(\vec{A}\), \(\vec{B}\), and \(\vec{R}\):

\[ R^2 = A^2 + B^2 + 2AB\cos(\beta) \]

Where \(\beta\) is the angle between \(\vec{A}\) and \(\vec{B}\).

### Step 4: Analyze the Angles
From the triangle formed by the vectors, we know that:

\[ \alpha + \omega + \beta = 180^\circ \]

This implies that if \(\alpha\) increases, \(\beta\) must decrease and vice versa.

### Step 5: Compare the Angles
If we assume that \(\alpha < \beta\), then according to the triangle inequality, the side opposite to the larger angle (which is \(\beta\)) must be greater than the side opposite to the smaller angle (which is \(\alpha\)). Therefore, we can conclude that:

\[ |\vec{B}| > |\vec{A}| \]

### Conclusion
Thus, the correct relationship derived from the angles and the sides is that:

\[ \alpha < \beta \implies |\vec{B}| > |\vec{A}| \]

### Final Answer
The resultant vector \(\vec{R}\) makes an angle \(\alpha\) with \(\vec{A}\) and an angle \(\omega\) with \(\vec{B}\) such that \(\alpha < \beta\) and \(|\vec{B}| > |\vec{A}|\).

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