The non-zero vectors are `vec a,vec b and vec c` are related by `vec a= 8vec b and vec c = -7vec b`. Then the angle between `vec a and vec c` is

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. 0

Explanation

To find the angle between the vectors \( \vec{a} \) and \( \vec{c} \) given the relationships \( \vec{a} = 8\vec{b} \) and \( \vec{c} = -7\vec{b} \), we can follow these steps:

### Step 1: Write the expressions for \( \vec{a} \) and \( \vec{c} \)
Given:
\[ \vec{a} = 8\vec{b} \]
\[ \vec{c} = -7\vec{b} \]

### Step 2: Use the dot product to find the angle between \( \vec{a} \) and \( \vec{c} \)
The dot product formula states:
\[ \vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos \theta \]
where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{c} \).

### Step 3: Calculate \( \vec{a} \cdot \vec{c} \)
Substituting the expressions for \( \vec{a} \) and \( \vec{c} \):
\[ \vec{a} \cdot \vec{c} = (8\vec{b}) \cdot (-7\vec{b}) = -56 (\vec{b} \cdot \vec{b}) = -56 |\vec{b}|^2 \]

### Step 4: Calculate the magnitudes of \( \vec{a} \) and \( \vec{c} \)
The magnitudes are:
\[ |\vec{a}| = |8\vec{b}| = 8|\vec{b}| \]
\[ |\vec{c}| = |-7\vec{b}| = 7|\vec{b}| \]

### Step 5: Substitute the magnitudes into the dot product formula
Now substituting back into the dot product equation:
\[ -56 |\vec{b}|^2 = (8|\vec{b}|)(7|\vec{b}|) \cos \theta \]
This simplifies to:
\[ -56 |\vec{b}|^2 = 56 |\vec{b}|^2 \cos \theta \]

### Step 6: Divide both sides by \( |\vec{b}|^2 \) (assuming \( |\vec{b}| \neq 0 \))
\[ -56 = 56 \cos \theta \]

### Step 7: Solve for \( \cos \theta \)
Dividing both sides by 56 gives:
\[ \cos \theta = -1 \]

### Step 8: Find the angle \( \theta \)
The angle \( \theta \) for which \( \cos \theta = -1 \) is:
\[ \theta = \pi \text{ radians} \]

### Final Answer
The angle between \( \vec{a} \) and \( \vec{c} \) is \( \pi \) radians.

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