If a,b,c be non-zero vectors such that a is perpendicular to b and c and `|a|=1,|b|=2,|c|=1,b*c=1` and there is a non-zero vector d coplanar with a+b and 2b-c and `d*a=1`, then minimum value of |d| is

Options

A. `(2)/(sqrt(13))`
B. `(3)/(sqrt(13))`
C. `(4)/(sqrt(5))`
D. `(4)/(sqrt(13))`.

Answer

To solve the problem step by step, we will break down the information given and use vector properties to find the minimum value of the magnitude of vector \( d \).

### Step 1: Understand the Given Information
We have three vectors \( a, b, c \) with the following properties:
- \( a \) is perpendicular to both \( b \) and \( c \).
- \( |a| = 1 \)
- \( |b| = 2 \)
- \( |c| = 1 \)
- \( b \cdot c = 1 \)
- There exists a non-zero vector \( d \) coplanar with \( a + b \) and \( 2b - c \).
- \( d \cdot a = 1 \)

### Step 2: Express \( d \) in Terms of \( a, b, c \)
Since \( d \) is coplanar with \( a + b \) and \( 2b - c \), we can express \( d \) as a linear combination of these vectors:
\[ d = \lambda (a + b) + \mu (2b - c) \]
for some scalars \( \lambda \) and \( \mu \).

### Step 3: Use the Condition \( d \cdot a = 1 \)
Substituting the expression for \( d \) into the condition \( d \cdot a = 1 \):
\[ d \cdot a = \lambda (a + b) \cdot a + \mu (2b - c) \cdot a \]
Since \( a \) is perpendicular to \( b \) and \( c \):
\[ = \lambda (a \cdot a) + \mu (2b \cdot a - c \cdot a) \]
This simplifies to:
\[ = \lambda (1) + \mu (0) = \lambda \]
Thus, we have:
\[ \lambda = 1 \]

### Step 4: Substitute \( \lambda \) Back into \( d \)
Now substituting \( \lambda = 1 \) into the expression for \( d \):
\[ d = (a + b) + \mu (2b - c) = a + b + 2\mu b - \mu c \]
This simplifies to:
\[ d = a + (1 + 2\mu)b - \mu c \]

### Step 5: Calculate the Magnitude of \( d \)
Now we calculate the magnitude \( |d|^2 \):
\[ |d|^2 = |a|^2 + |(1 + 2\mu)b|^2 + |-\mu c|^2 + 2((1 + 2\mu)b) \cdot a + 2(-\mu c) \cdot a \]
Substituting the known magnitudes:
\[ = 1 + (1 + 2\mu)^2 |b|^2 + \mu^2 |c|^2 + 0 + 0 \]
Since \( |b| = 2 \) and \( |c| = 1 \):
\[ = 1 + (1 + 2\mu)^2 \cdot 4 + \mu^2 \]
Expanding this:
\[ = 1 + 4(1 + 4\mu + 4\mu^2) + \mu^2 \]
\[ = 1 + 4 + 16\mu + 16\mu^2 + \mu^2 = 5 + 16\mu + 17\mu^2 \]

### Step 6: Minimize \( |d|^2 \)
To find the minimum value of \( |d|^2 \), we can treat it as a quadratic in \( \mu \):
\[ |d|^2 = 17\mu^2 + 16\mu + 5 \]
The minimum occurs at:
\[ \mu = -\frac{b}{2a} = -\frac{16}{2 \cdot 17} = -\frac{8}{17} \]

### Step 7: Substitute \( \mu \) Back to Find Minimum Value
Substituting \( \mu = -\frac{8}{17} \) back into the equation for \( |d|^2 \):
\[ |d|^2 = 17\left(-\frac{8}{17}\right)^2 + 16\left(-\frac{8}{17}\right) + 5 \]
Calculating:
\[ = 17 \cdot \frac{64}{289} - \frac{128}{17} + 5 \]
\[ = \frac{1088}{289} - \frac{2176}{289} + \frac{1445}{289} \]
\[ = \frac{1088 - 2176 + 1445}{289} = \frac{357}{289} \]

### Step 8: Find the Minimum Magnitude \( |d| \)
Thus, the minimum value of \( |d| \) is:
\[ |d| = \sqrt{\frac{357}{289}} = \frac{\sqrt{357}}{17} \]

### Final Answer
The minimum value of \( |d| \) is \( \frac{\sqrt{357}}{17} \).