If the angles between the vectors `veca` and `vecb, vecb and vecc, vecc` and `veca` be `(pi)/(4),(pi)/(3),(pi)/(3)` respectively, then the angle which `veca` makes with the plane containing `vecb and vecc` is

Options

A. `sin^(-1)sqrt((sqrt2)/(3))`
B. `sin^(-1).(2)/(3)`
C. `sin^(-1).(1)/(4)`
D. `sin^(-1)sqrt((2)/(3))`

Answer

To find the angle that vector \(\vec{a}\) makes with the plane containing vectors \(\vec{b}\) and \(\vec{c}\), we can follow these steps:

### Step 1: Understand the relationship between the vectors and their angles
Given:
- The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\).
- The angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\).
- The angle between \(\vec{c}\) and \(\vec{a}\) is \(\frac{\pi}{3}\).

### Step 2: Use the dot product to express relationships
Using the cosine of the angles, we can write:
1. \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} |\vec{a}| |\vec{b}|\)
2. \(\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} |\vec{b}| |\vec{c}|\)
3. \(\vec{c} \cdot \vec{a} = |\vec{c}| |\vec{a}| \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} |\vec{c}| |\vec{a}|\)

### Step 3: Find the sine of the angle between \(\vec{b}\) and \(\vec{c}\)
The sine of the angle between \(\vec{b}\) and \(\vec{c}\) is given by:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin\left(\frac{\pi}{3}\right) = |\vec{b}| |\vec{c}| \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Use the normal vector to find the angle \(\theta\)
The normal vector to the plane formed by \(\vec{b}\) and \(\vec{c}\) is \(\vec{b} \times \vec{c}\). The angle \(\theta\) that \(\vec{a}\) makes with this plane can be determined using the formula:
\[ \sin \theta = \frac{|\vec{a} \cdot (\vec{b} \times \vec{c})|}{|\vec{a}| |\vec{b} \times \vec{c}|} \]

### Step 5: Calculate the scalar triple product
The scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) can be expressed as:
\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = \text{det}(\vec{a}, \vec{b}, \vec{c}) \]
This can be calculated using the relationships established earlier.

### Step 6: Substitute and simplify
Substituting the values:
\[ \sin \theta = \frac{|\vec{a}| \cdot |\vec{b}| \cdot |\vec{c}| \cdot \frac{\sqrt{3}}{2}}{|\vec{a}| \cdot |\vec{b}| \cdot |\vec{c}|} = \frac{\sqrt{3}}{2} \]

### Step 7: Find the angle
Thus, we can find \(\theta\):
\[ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]

### Final Answer
The angle which \(\vec{a}\) makes with the plane containing \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\).

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