The points O, A, B, C, D are such that `vecOA = a, vecOB = b, vecOC = 2a + 3b and vecOD = a-2b`. If `a =3b`. Then the angle between `vecBD and vecAC` is .......

Answer

To find the angle between the vectors \(\vec{BD}\) and \(\vec{AC}\), we will follow these steps:

### Step 1: Define the vectors
Given:
- \(\vec{OA} = \vec{a}\)
- \(\vec{OB} = \vec{b}\)
- \(\vec{OC} = 2\vec{a} + 3\vec{b}\)
- \(\vec{OD} = \vec{a} - 2\vec{b}\)

We also know that \(\vec{a} = 3\vec{b}\).

### Step 2: Substitute \(\vec{a}\) in terms of \(\vec{b}\)
Since \(\vec{a} = 3\vec{b}\), we can substitute this into the expressions for \(\vec{OC}\) and \(\vec{OD}\):
- \(\vec{OC} = 2(3\vec{b}) + 3\vec{b} = 6\vec{b} + 3\vec{b} = 9\vec{b}\)
- \(\vec{OD} = 3\vec{b} - 2\vec{b} = \vec{b}\)

### Step 3: Calculate \(\vec{BD}\) and \(\vec{AC}\)
Now, we can find the vectors \(\vec{BD}\) and \(\vec{AC}\):
- \(\vec{BD} = \vec{OD} - \vec{OB} = \vec{b} - \vec{b} = \vec{0}\)
- \(\vec{AC} = \vec{OC} - \vec{OA} = 9\vec{b} - 3\vec{b} = 6\vec{b}\)

### Step 4: Find the angle between \(\vec{BD}\) and \(\vec{AC}\)
To find the angle \(\theta\) between the vectors \(\vec{BD}\) and \(\vec{AC}\), we use the dot product formula:
\[ \vec{BD} \cdot \vec{AC} = |\vec{BD}| |\vec{AC}| \cos(\theta) \]

Since \(\vec{BD} = \vec{0}\), we have:
\[ \vec{0} \cdot \vec{AC} = 0 \]
This implies:
\[ |\vec{0}| |\vec{AC}| \cos(\theta) = 0 \]

### Step 5: Conclusion about the angle
Since the dot product is zero, it means that the angle \(\theta\) can be any angle, but since \(\vec{BD}\) is the zero vector, it does not have a defined direction. Therefore, we conclude that the angle between \(\vec{BD}\) and \(\vec{AC}\) is undefined.