The non zero vectors `veca,vecb, and vecc ` are related byi `veca=8vecb nd vecc=-7vecb.` Then the angle between `veca and vecc` is (A) `pi` (B) 0 (C) `pi/4` (D) `pi/2`

Options

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

Explanation

To find the angle between the vectors \(\vec{a}\) and \(\vec{c}\), we can follow these steps:

### Step 1: Write down the relationships between the vectors
Given:
\[ \vec{a} = 8\vec{b} \]
\[ \vec{c} = -7\vec{b} \]

### Step 2: Understand the direction of the vectors
Both vectors \(\vec{a}\) and \(\vec{c}\) are expressed in terms of \(\vec{b}\). The vector \(\vec{a}\) is a scalar multiple of \(\vec{b}\) in the positive direction, while \(\vec{c}\) is a scalar multiple of \(\vec{b}\) in the negative direction.

### Step 3: Determine the relationship between \(\vec{a}\) and \(\vec{c}\)
Since \(\vec{c} = -7\vec{b}\), we can rewrite \(\vec{c}\) as:
\[ \vec{c} = -7\vec{b} = 7(-\vec{b}) \]
This indicates that \(\vec{c}\) is in the opposite direction to \(\vec{b}\).

### Step 4: Analyze the angle between \(\vec{a}\) and \(\vec{c}\)
Since \(\vec{a}\) and \(\vec{c}\) are both collinear and point in opposite directions, the angle between them is:
\[ \theta = \pi \text{ radians} \]

### Conclusion
Thus, the angle between \(\vec{a}\) and \(\vec{c}\) is \(\pi\) radians.

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

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