If `vec(A) = 3hat(i) - 4hat(j) + hat(k)` and `vec(B) = 4hat(j) + phat(i) + hat(k)` for what value of p, `vec(A)` and `vec(B)` will ve collinear ?

Options

A. 3
B. `-5`
C. `-(16)/(3)`
D. `vec(A)` and `vec(B)` cannot be collinear

Answer

To determine the value of \( p \) for which the vectors \( \vec{A} \) and \( \vec{B} \) are collinear, we start by writing down the vectors:

\[ \vec{A} = 3\hat{i} - 4\hat{j} + \hat{k} \]
\[ \vec{B} = p\hat{i} + 4\hat{j} + \hat{k} \]

### Step 1: Identify the direction ratios of the vectors
The direction ratios of vector \( \vec{A} \) are \( (3, -4, 1) \) and for vector \( \vec{B} \) they are \( (p, 4, 1) \).

### Step 2: Set up the proportionality condition
For the vectors to be collinear, their direction ratios must be proportional. This means we can set up the following equations based on the ratios:

\[ \frac{3}{p} = \frac{-4}{4} = \frac{1}{1} \]

### Step 3: Solve the first ratio
From the second part of the equation, we have:

\[ \frac{-4}{4} = -1 \]

This means:

\[ \frac{3}{p} = -1 \]

Cross-multiplying gives:

\[ 3 = -p \]

Thus, we find:

\[ p = -3 \]

### Step 4: Check the third ratio
Now, we verify the third part of the ratio:

\[ \frac{3}{p} = \frac{1}{1} \]

Substituting \( p = -3 \):

\[ \frac{3}{-3} = -1 \]

This is consistent with our earlier calculations.

### Conclusion
Thus, the value of \( p \) for which the vectors \( \vec{A} \) and \( \vec{B} \) are collinear is:

\[ \boxed{-3} \]