Let `veca=2hati+2hatj+hatk` and `vecc` is a vector such that `|vecaxxvecc|^(2)+(veca.vecc)^(2)=144` then `|vecc|` is equal to

Options

A. 16
B. 4
C. 3
D. 9

Answer

To solve the problem step by step, we will follow the reasoning and calculations laid out in the video transcript.

### Step 1: Identify the vector \(\vec{a}\)
Given:
\[ \vec{a} = 2\hat{i} + 2\hat{j} + \hat{k} \]

### Step 2: Calculate the magnitude of vector \(\vec{a}\)
The magnitude of vector \(\vec{a}\) is calculated as follows:
\[ |\vec{a}| = \sqrt{(2)^2 + (2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 3: Use the given condition
We are given the condition:
\[ |\vec{a} \times \vec{c}|^2 + (\vec{a} \cdot \vec{c})^2 = 144 \]

### Step 4: Express \(|\vec{a} \times \vec{c}|\) and \(\vec{a} \cdot \vec{c}\)
Let \(|\vec{c}| = c\) and \(\theta\) be the angle between \(\vec{a}\) and \(\vec{c}\).

Using the formulas for cross and dot products:
\[ |\vec{a} \times \vec{c}| = |\vec{a}| |\vec{c}| \sin \theta = 3c \sin \theta \]
\[ \vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos \theta = 3c \cos \theta \]

### Step 5: Substitute into the given condition
Substituting these expressions into the given condition:
\[ (3c \sin \theta)^2 + (3c \cos \theta)^2 = 144 \]
This simplifies to:
\[ 9c^2 \sin^2 \theta + 9c^2 \cos^2 \theta = 144 \]

### Step 6: Factor out common terms
Factoring out \(9c^2\):
\[ 9c^2 (\sin^2 \theta + \cos^2 \theta) = 144 \]

### Step 7: Use the Pythagorean identity
Since \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ 9c^2 \cdot 1 = 144 \]
Thus:
\[ 9c^2 = 144 \]

### Step 8: Solve for \(|\vec{c}|\)
Dividing both sides by 9:
\[ c^2 = \frac{144}{9} = 16 \]
Taking the square root:
\[ c = 4 \]

### Conclusion
The magnitude of vector \(\vec{c}\) is:
\[ |\vec{c}| = 4 \]