Let `bara,barb and barc` be non-zero vectors such that `(bara times barb)times barc=1/3|barb||barc|bara`. If `theta` is the obtuse angle between the vectors `barb and barc` then `sin theta` equals

Options

A. `1/3`
B. `sqrt2/3`
C. `2/3`
D. `(2sqrt2)/3`

Answer

To solve the given problem step by step, we will use the properties of vector products and some trigonometric identities.

### Step 1: Understand the given condition
We are given that:
\[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{3} |\mathbf{b}| |\mathbf{c}| \mathbf{a} \]
This is a vector equation involving the cross product of vectors.

### Step 2: Use the vector triple product identity
We can use the vector triple product identity:
\[ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} \]
Applying this to our equation, we have:
\[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

### Step 3: Set the two sides equal
Now we can set the left-hand side equal to the right-hand side:
\[ (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} = \frac{1}{3} |\mathbf{b}| |\mathbf{c}| \mathbf{a} \]

### Step 4: Compare coefficients
For the vectors to be equal, their coefficients must also be equal. This gives us two equations:
1. \(\mathbf{a} \cdot \mathbf{c} = \frac{1}{3} |\mathbf{b}| |\mathbf{c}|\)
2. \(-(\mathbf{a} \cdot \mathbf{b}) = 0\)

From the second equation, we conclude:
\[ \mathbf{a} \cdot \mathbf{b} = 0 \]
This means that vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal.

### Step 5: Analyze the first equation
From the first equation, we have:
\[ \mathbf{a} \cdot \mathbf{c} = \frac{1}{3} |\mathbf{b}| |\mathbf{c}| \]

### Step 6: Relate to cosine of the angle
Using the definition of the dot product:
\[ \mathbf{a} \cdot \mathbf{c} = |\mathbf{a}| |\mathbf{c}| \cos \theta \]
where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{c}\). Thus, we can write:
\[ |\mathbf{a}| |\mathbf{c}| \cos \theta = \frac{1}{3} |\mathbf{b}| |\mathbf{c}| \]

### Step 7: Simplify the equation
Assuming \(|\mathbf{c}| \neq 0\), we can divide both sides by \(|\mathbf{c}|\):
\[ |\mathbf{a}| \cos \theta = \frac{1}{3} |\mathbf{b}| \]

### Step 8: Use the cosine of the angle between \(\mathbf{b}\) and \(\mathbf{c}\)
Now, we know that:
\[ \mathbf{b} \cdot \mathbf{c} = |\mathbf{b}| |\mathbf{c}| \cos \phi \]
where \(\phi\) is the angle between \(\mathbf{b}\) and \(\mathbf{c}\). Given that \(\mathbf{b} \cdot \mathbf{c} = -\frac{1}{3} |\mathbf{b}| |\mathbf{c}|\), we have:
\[ |\mathbf{b}| |\mathbf{c}| \cos \phi = -\frac{1}{3} |\mathbf{b}| |\mathbf{c}| \]
Thus,
\[ \cos \phi = -\frac{1}{3} \]

### Step 9: Find \(\sin \phi\)
Using the identity \(\sin^2 \phi + \cos^2 \phi = 1\):
\[ \sin^2 \phi = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]
Thus,
\[ \sin \phi = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \]

### Final Answer
Therefore, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{2\sqrt{2}}{3} \]

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