The resultant of two vectors is perpendicular to first vector of magnitude 6N. If the resultant has magnitude `6sqrt(3)N`, then magnitude of second vector is

Options

A. `6sqrt(2)N`
B. `12N`
C. `9sqrt(3)N`
D. `6sqrt(3)N`

Answer

To solve the problem step by step, we will use vector resolution and the properties of right triangles.

### Step 1: Understand the Problem
We have two vectors:
- Vector A with a magnitude of 6 N.
- The resultant vector R, which is perpendicular to vector A and has a magnitude of \(6\sqrt{3}\) N.

We need to find the magnitude of the second vector, which we will call vector B.

### Step 2: Set Up the Equations
Since the resultant vector R is perpendicular to vector A, we can use the Pythagorean theorem. The relationship can be expressed as:
\[ R^2 = A^2 + B^2 \]
Where:
- \(R = 6\sqrt{3}\) N
- \(A = 6\) N
- \(B\) is the magnitude of the second vector we need to find.

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ (6\sqrt{3})^2 = (6)^2 + B^2 \]
Calculating the squares:
\[ 108 = 36 + B^2 \]

### Step 4: Solve for B^2
Now, isolate \(B^2\):
\[ B^2 = 108 - 36 \]
\[ B^2 = 72 \]

### Step 5: Find B
Taking the square root of both sides:
\[ B = \sqrt{72} \]
\[ B = \sqrt{36 \times 2} = 6\sqrt{2} \text{ N} \]

### Conclusion
The magnitude of the second vector B is \(6\sqrt{2}\) N.

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