Given that `vecu = hati + 2hatj + 3hatk , vecv = 2hati + hatk + 4hatk , vecw = hati + 3hatj + 3hatk and (vecu.vecR - 15) hati + (vecc. vecR - 30) hatj + (vecw . vec- 20) veck = vec0`. Then find the greatest integer less than or equal to `|vecR|`.

Explanation

To solve the given problem step by step, we will follow the instructions provided in the video transcript and derive the solution systematically.

### Step 1: Define the vectors
We are given:
- \(\vec{u} = \hat{i} + 2\hat{j} + 3\hat{k}\)
- \(\vec{v} = 2\hat{i} + \hat{j} + 4\hat{k}\)
- \(\vec{w} = \hat{i} + 3\hat{j} + 3\hat{k}\)

### Step 2: Write the equation
The problem states:
\[ (\vec{u} \cdot \vec{R} - 15) \hat{i} + (\vec{v} \cdot \vec{R} - 30) \hat{j} + (\vec{w} \cdot \vec{R} - 20) \hat{k} = \vec{0} \]
This implies:
\[ \vec{u} \cdot \vec{R} - 15 = 0 \]
\[ \vec{v} \cdot \vec{R} - 30 = 0 \]
\[ \vec{w} \cdot \vec{R} - 20 = 0 \]

### Step 3: Assume \(\vec{R}\)
Let \(\vec{R} = x\hat{i} + y\hat{j} + z\hat{k}\).

### Step 4: Set up the dot product equations
From the equations derived:
1. \(\vec{u} \cdot \vec{R} = 15\)
\[ (1)(x) + (2)(y) + (3)(z) = 15 \quad \text{(1)} \]
2. \(\vec{v} \cdot \vec{R} = 30\)
\[ (2)(x) + (1)(y) + (4)(z) = 30 \quad \text{(2)} \]
3. \(\vec{w} \cdot \vec{R} = 20\)
\[ (1)(x) + (3)(y) + (3)(z) = 20 \quad \text{(3)} \]

### Step 5: Solve the system of equations
We now have a system of three equations:
1. \(x + 2y + 3z = 15\)
2. \(2x + y + 4z = 30\)
3. \(x + 3y + 3z = 20\)

We can solve this system using substitution or elimination.

From (1):
\[ x = 15 - 2y - 3z \quad \text{(4)} \]

Substituting (4) into (2):
\[ 2(15 - 2y - 3z) + y + 4z = 30 \]
\[ 30 - 4y - 6z + y + 4z = 30 \]
\[ -3y - 2z = 0 \implies 3y + 2z = 0 \quad \text{(5)} \]

Now substituting (4) into (3):
\[ (15 - 2y - 3z) + 3y + 3z = 20 \]
\[ 15 - 2y - 3z + 3y + 3z = 20 \]
\[ 15 + y = 20 \implies y = 5 \quad \text{(6)} \]

Substituting (6) back into (5):
\[ 3(5) + 2z = 0 \implies 15 + 2z = 0 \implies z = -7.5 \quad \text{(7)} \]

Now substituting (6) and (7) back into (4):
\[ x = 15 - 2(5) - 3(-7.5) = 15 - 10 + 22.5 = 27.5 \quad \text{(8)} \]

### Step 6: Calculate the magnitude of \(\vec{R}\)
Now we have:
\[ x = 27.5, \quad y = 5, \quad z = -7.5 \]
Thus,
\[ \vec{R} = 27.5\hat{i} + 5\hat{j} - 7.5\hat{k} \]

The magnitude of \(\vec{R}\) is given by:
\[ |\vec{R}| = \sqrt{x^2 + y^2 + z^2} = \sqrt{(27.5)^2 + (5)^2 + (-7.5)^2} \]
Calculating:
\[ |\vec{R}| = \sqrt{756.25 + 25 + 56.25} = \sqrt{837.5} \approx 28.96 \]

### Step 7: Find the greatest integer less than or equal to \(|\vec{R}|\)
The greatest integer less than or equal to \(28.96\) is \(28\).

### Final Answer
The greatest integer less than or equal to \(|\vec{R}|\) is \(28\).

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