Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p x ((x-q) x p) + q x ((x-r) x q) + r x ((x-p) x r)=0 Then x is given by :

Answer

To solve the problem, we start with the equation given in the question:

\[ p \times x + (x - q) \times p + q \times (x - r) \times q + r \times (x - p) \times r = 0 \]

### Step 1: Rewrite the equation
We can rewrite the equation as follows:

\[ p \times x + (x \times p - q \times p) + (q \times x - r \times q) + (r \times x - p \times r) = 0 \]

### Step 2: Combine like terms
Now, we can combine the terms involving \(x\):

\[ (p \times x + x \times p + q \times x + x \times q + r \times x + x \times r) - (q \times p + r \times q + p \times r) = 0 \]

### Step 3: Factor out \(x\)
Notice that \(p \times x + x \times p\) can be simplified, as can the other terms. Thus, we can factor out \(x\):

\[ (p + q + r) \times x - (q \times p + r \times q + p \times r) = 0 \]

### Step 4: Solve for \(x\)
From the equation above, we can isolate \(x\):

\[ (p + q + r) \times x = q \times p + r \times q + p \times r \]

### Step 5: Express \(x\)
Now, we can express \(x\):

\[ x = \frac{(q \times p + r \times q + p \times r)}{(p + q + r)} \]

### Step 6: Recognize the vectors are mutually perpendicular
Since \(p\), \(q\), and \(r\) are mutually perpendicular vectors of the same magnitude, we can denote their magnitudes as \(k\). Thus, we have:

\[ |p| = |q| = |r| = k \]

### Step 7: Final expression for \(x\)
Given that \(p\), \(q\), and \(r\) are mutually perpendicular, we can conclude that:

\[ x = p + q + r \]

### Final Answer
Thus, the vector \(x\) is given by:

\[ x = p + q + r \]

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