If the vectors `vec a =2hati + 3hatj +6hatk and vec b` are collinear and ` |vec b |=21, " then " vec b=`
(A) `pm 3(2hati + 3 hatj + 6 hatk)`
(B) `pm (2hati + 3hatj - 6 hatk)`
(C)`pm 21(2hati + 3 hatj + 6 hatk)`
(D)`pm 21(hati + hatj + hatk)`

Options

A. `pm 3(2hati + 3 hatj + 6 hatk)`
B. `pm (2hati + 3hatj - 6 hatk)`
C. `pm 21(2hati + 3 hatj + 6 hatk)`
D. `pm 21(hati + hatj + hatk)`

Answer

To solve the problem, we need to determine the vector \(\vec{b}\) given that it is collinear with \(\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}\) and has a magnitude of 21.

### Step 1: Understand the relationship between collinear vectors
If two vectors are collinear, one can be expressed as a scalar multiple of the other. Therefore, we can write:
\[ \vec{b} = \lambda \vec{a} \]
for some scalar \(\lambda\).

### Step 2: Calculate the magnitude of vector \(\vec{a}\)
The magnitude of vector \(\vec{a}\) is calculated using the formula:
\[ |\vec{a}| = \sqrt{(a_x)^2 + (a_y)^2 + (a_z)^2} \]
Substituting the components of \(\vec{a}\):
\[ |\vec{a}| = \sqrt{(2)^2 + (3)^2 + (6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 3: Relate the magnitudes of \(\vec{b}\) and \(\vec{a}\)
We know that the magnitude of \(\vec{b}\) is given as 21. Therefore, we can set up the equation:
\[ |\vec{b}| = |\lambda \vec{a}| = |\lambda| |\vec{a}| \]
Substituting the known values:
\[ 21 = |\lambda| \cdot 7 \]

### Step 4: Solve for \(|\lambda|\)
To find \(|\lambda|\), we rearrange the equation:
\[ |\lambda| = \frac{21}{7} = 3 \]
Thus, \(\lambda\) can be either \(3\) or \(-3\), giving us:
\[ \lambda = \pm 3 \]

### Step 5: Substitute \(\lambda\) back to find \(\vec{b}\)
Now we substitute \(\lambda\) back into the expression for \(\vec{b}\):
\[ \vec{b} = \lambda \vec{a} = \pm 3(2\hat{i} + 3\hat{j} + 6\hat{k}) \]
This simplifies to:
\[ \vec{b} = \pm (6\hat{i} + 9\hat{j} + 18\hat{k}) \]

### Step 6: Check the options
We need to match this result with the given options. The expression \(6\hat{i} + 9\hat{j} + 18\hat{k}\) can be factored as:
\[ \vec{b} = \pm 3(2\hat{i} + 3\hat{j} + 6\hat{k}) \]
This matches option (A).

### Final Answer
Thus, the correct answer is:
\[ \vec{b} = \pm 3(2\hat{i} + 3\hat{j} + 6\hat{k}) \]

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