If `(a+3b)*(7a-5b)=0 and (a-4b)*(7a-2b)=0`. Then, the angle between a and b is

Options

A. `60^(@)`
B. `30^(@)`
C. `90^(@)`
D. none of these

Answer

To solve the problem step by step, we will analyze the given conditions and derive the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

### Step 1: Analyze the first condition
We start with the first condition:
\[ (\mathbf{a} + 3\mathbf{b}) \cdot (7\mathbf{a} - 5\mathbf{b}) = 0 \]
Expanding this using the distributive property of the dot product:
\[ \mathbf{a} \cdot (7\mathbf{a}) + \mathbf{a} \cdot (-5\mathbf{b}) + 3\mathbf{b} \cdot (7\mathbf{a}) + 3\mathbf{b} \cdot (-5\mathbf{b}) = 0 \]
This simplifies to:
\[ 7 \|\mathbf{a}\|^2 - 5 \mathbf{a} \cdot \mathbf{b} + 21 \mathbf{b} \cdot \mathbf{a} - 15 \|\mathbf{b}\|^2 = 0 \]
Combining like terms gives:
\[ 7 \|\mathbf{a}\|^2 + 16 \mathbf{a} \cdot \mathbf{b} - 15 \|\mathbf{b}\|^2 = 0 \tag{1} \]

### Step 2: Analyze the second condition
Now, we analyze the second condition:
\[ (\mathbf{a} - 4\mathbf{b}) \cdot (7\mathbf{a} - 2\mathbf{b}) = 0 \]
Expanding this:
\[ \mathbf{a} \cdot (7\mathbf{a}) + \mathbf{a} \cdot (-2\mathbf{b}) - 4\mathbf{b} \cdot (7\mathbf{a}) + 4\mathbf{b} \cdot (2\mathbf{b}) = 0 \]
This simplifies to:
\[ 7 \|\mathbf{a}\|^2 - 2 \mathbf{a} \cdot \mathbf{b} - 28 \mathbf{b} \cdot \mathbf{a} + 8 \|\mathbf{b}\|^2 = 0 \]
Combining like terms gives:
\[ 7 \|\mathbf{a}\|^2 - 30 \mathbf{a} \cdot \mathbf{b} + 8 \|\mathbf{b}\|^2 = 0 \tag{2} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( 7 \|\mathbf{a}\|^2 + 16 \mathbf{a} \cdot \mathbf{b} - 15 \|\mathbf{b}\|^2 = 0 \)
2. \( 7 \|\mathbf{a}\|^2 - 30 \mathbf{a} \cdot \mathbf{b} + 8 \|\mathbf{b}\|^2 = 0 \)

### Step 4: Subtract the equations
Subtract equation (2) from equation (1):
\[ (7 \|\mathbf{a}\|^2 + 16 \mathbf{a} \cdot \mathbf{b} - 15 \|\mathbf{b}\|^2) - (7 \|\mathbf{a}\|^2 - 30 \mathbf{a} \cdot \mathbf{b} + 8 \|\mathbf{b}\|^2) = 0 \]
This simplifies to:
\[ 46 \mathbf{a} \cdot \mathbf{b} - 23 \|\mathbf{b}\|^2 = 0 \]
From this, we can derive:
\[ \|\mathbf{b}\|^2 = 2 \mathbf{a} \cdot \mathbf{b} \tag{3} \]

### Step 5: Multiply and rearrange
Next, we multiply equation (1) by 8:
\[ 56 \|\mathbf{a}\|^2 + 128 \mathbf{a} \cdot \mathbf{b} - 120 \|\mathbf{b}\|^2 = 0 \tag{4} \]
And multiply equation (2) by 15:
\[ 105 \|\mathbf{a}\|^2 - 450 \mathbf{a} \cdot \mathbf{b} + 120 \|\mathbf{b}\|^2 = 0 \tag{5} \]

### Step 6: Add the equations
Now, add equations (4) and (5):
\[ (105 + 56) \|\mathbf{a}\|^2 + (-450 + 128) \mathbf{a} \cdot \mathbf{b} + (120 - 120) \|\mathbf{b}\|^2 = 0 \]
This simplifies to:
\[ 161 \|\mathbf{a}\|^2 - 322 \mathbf{a} \cdot \mathbf{b} = 0 \]
From this, we can derive:
\[ \|\mathbf{a}\|^2 = 2 \mathbf{a} \cdot \mathbf{b} \tag{4} \]

### Step 7: Relate the equations
Now we have two equations:
1. \( \|\mathbf{b}\|^2 = 2 \mathbf{a} \cdot \mathbf{b} \)
2. \( \|\mathbf{a}\|^2 = 2 \mathbf{a} \cdot \mathbf{b} \)

### Step 8: Substitute and solve for angle
Substituting \( \|\mathbf{b}\|^2 \) from (3) into (4):
\[ \|\mathbf{b}\|^2 = \|\mathbf{a}\|^2 \]
This implies:
\[ \|\mathbf{a}\| = \|\mathbf{b}\| \]
Using the cosine formula:
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \]
Substituting \( \|\mathbf{b}\| = \|\mathbf{a}\| \):
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|^2} \]
From our earlier equations, we can derive:
\[ \mathbf{a} \cdot \mathbf{b} = \frac{1}{4} \|\mathbf{a}\|^2 \]
Thus:
\[ \cos \theta = \frac{1/4 \|\mathbf{a}\|^2}{\|\mathbf{a}\|^2} = \frac{1}{4} \]
This gives:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Final Answer
The angle between \( \mathbf{a} \) and \( \mathbf{b} \) is \( 60^\circ \).

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