The non-zero vectors `bara,barb` and `barc` are related by `bara=8barb` and `barc=-7barb`. Then, the angle between `bara` and `barc` is of measures

Options

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

Answer

To find the angle between the vectors \(\bar{a}\) and \(\bar{c}\), we start with the given relationships:

\[ \bar{a} = 8\bar{b} \]
\[ \bar{c} = -7\bar{b} \]

### Step 1: Use the dot product formula

The angle \(\theta\) between two vectors can be found using the dot product:

\[ \bar{a} \cdot \bar{c} = |\bar{a}| |\bar{c}| \cos \theta \]

### Step 2: Substitute the expressions for \(\bar{a}\) and \(\bar{c}\)

Substituting the expressions for \(\bar{a}\) and \(\bar{c}\):

\[ \bar{a} \cdot \bar{c} = (8\bar{b}) \cdot (-7\bar{b}) \]

### Step 3: Calculate the dot product

Using the properties of the dot product:

\[ \bar{a} \cdot \bar{c} = 8 \cdot (-7) (\bar{b} \cdot \bar{b}) = -56 |\bar{b}|^2 \]

### Step 4: Calculate the magnitudes of \(\bar{a}\) and \(\bar{c}\)

Now, we calculate the magnitudes:

\[ |\bar{a}| = |8\bar{b}| = 8 |\bar{b}| \]
\[ |\bar{c}| = |-7\bar{b}| = 7 |\bar{b}| \]

### Step 5: Substitute the magnitudes into the dot product equation

Now substituting back into the dot product formula:

\[ -56 |\bar{b}|^2 = (8 |\bar{b}|)(7 |\bar{b}|) \cos \theta \]

This simplifies to:

\[ -56 |\bar{b}|^2 = 56 |\bar{b}|^2 \cos \theta \]

### Step 6: Divide both sides by \( |\bar{b}|^2 \) (assuming \( |\bar{b}| \neq 0 \))

\[ -56 = 56 \cos \theta \]

### 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:

\[ \theta = \pi \]

### Conclusion

Thus, the angle between \(\bar{a}\) and \(\bar{c}\) is:

\[ \theta = \pi \]