The non-zero vectors a, b and c are related as b = 5a and c = -2b. The angle between a and c is

Options

A. `pi/2`
B. `pi/4`
C. `pi`
D. `pi/3`

Answer

To find the angle between the vectors \( \mathbf{a} \) and \( \mathbf{c} \), we can use the relationships given in the problem. Let's go through the solution step by step.

### Step 1: Identify the relationships between the vectors
We are given:
\[ \mathbf{b} = 5\mathbf{a} \]
\[ \mathbf{c} = -2\mathbf{b} \]

### Step 2: Substitute the expression for \(\mathbf{b}\) into the expression for \(\mathbf{c}\)
Substituting \(\mathbf{b}\) into the equation for \(\mathbf{c}\):
\[ \mathbf{c} = -2\mathbf{b} = -2(5\mathbf{a}) = -10\mathbf{a} \]

### Step 3: Use the dot product to find the angle between \(\mathbf{a}\) and \(\mathbf{c}\)
The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{c}\) can be expressed as:
\[ \mathbf{a} \cdot \mathbf{c} = |\mathbf{a}| |\mathbf{c}| \cos \theta \]
where \(\theta\) is the angle between the vectors \(\mathbf{a}\) and \(\mathbf{c}\).

### Step 4: Calculate \(|\mathbf{c}|\)
Using the expression we found for \(\mathbf{c}\):
\[ |\mathbf{c}| = |-10\mathbf{a}| = 10|\mathbf{a}| \]

### Step 5: Substitute into the dot product equation
Now substituting into the dot product equation:
\[ \mathbf{a} \cdot \mathbf{c} = \mathbf{a} \cdot (-10\mathbf{a}) = -10(\mathbf{a} \cdot \mathbf{a}) = -10|\mathbf{a}|^2 \]
Thus, we have:
\[ -10|\mathbf{a}|^2 = |\mathbf{a}|(10|\mathbf{a}|) \cos \theta \]

### Step 6: Simplify the equation
This simplifies to:
\[ -10|\mathbf{a}|^2 = 10|\mathbf{a}|^2 \cos \theta \]
Dividing both sides by \(10|\mathbf{a}|^2\) (since \(|\mathbf{a}|\) is non-zero):
\[ -1 = \cos \theta \]

### Step 7: Find the angle \(\theta\)
The angle \(\theta\) for which \(\cos \theta = -1\) is:
\[ \theta = \pi \text{ radians} \quad (\text{or } 180^\circ) \]

### Conclusion
The angle between vectors \(\mathbf{a}\) and \(\mathbf{c}\) is:
\[ \theta = \pi \text{ radians} \]