What is the value of p for which the vector `vec(p) (2 hat(i)- hat(j) +2hat(k))` is of 3 unit's length?

Options

A. 1
B. 2
C. 3
D. 6

Answer

To find the value of \( p \) for which the vector \( \vec{p} (2 \hat{i} - \hat{j} + 2 \hat{k}) \) has a length of 3 units, we can follow these steps:

### Step 1: Write the vector in terms of \( p \)
The vector can be expressed as:
\[ \vec{v} = p (2 \hat{i} - \hat{j} + 2 \hat{k}) = (2p) \hat{i} + (-p) \hat{j} + (2p) \hat{k} \]

### Step 2: Calculate the length of the vector
The length (magnitude) of a vector \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by:
\[ |\vec{A}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector \( \vec{v} \):
- \( a = 2p \)
- \( b = -p \)
- \( c = 2p \)

Thus, the length of the vector \( \vec{v} \) is:
\[ |\vec{v}| = \sqrt{(2p)^2 + (-p)^2 + (2p)^2} \]

### Step 3: Simplify the expression
Calculating the squares:
\[ |\vec{v}| = \sqrt{4p^2 + p^2 + 4p^2} = \sqrt{9p^2} \]
This simplifies to:
\[ |\vec{v}| = 3|p| \]

### Step 4: Set the length equal to 3 units
We want the length of the vector to be 3 units:
\[ 3|p| = 3 \]

### Step 5: Solve for \( p \)
Dividing both sides by 3:
\[ |p| = 1 \]
This gives us two possible solutions for \( p \):
\[ p = 1 \quad \text{or} \quad p = -1 \]

### Final Answer
The values of \( p \) for which the vector has a length of 3 units are \( p = 1 \) and \( p = -1 \).

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