Find the unit vector in the direction of vector ` vec(PQ)`, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Options

A. `hati/sqrt3+hatj/sqrt2+hatk/sqrt2`
B. `hati/sqrt2+hatj/sqrt3+hatk/sqrt3`
C. `hati/sqrt2+hatj/sqrt2+hatk/sqrt2`
D. `hati/sqrt3+hatj/sqrt3+hatk/sqrt3`

Explanation

To find the unit vector in the direction of the vector \( \vec{PQ} \), where \( P(1, 2, 3) \) and \( Q(4, 5, 6) \), we can follow these steps:

### Step 1: Find the vector \( \vec{PQ} \)

The vector \( \vec{PQ} \) can be calculated by subtracting the coordinates of point \( P \) from the coordinates of point \( Q \):

\[ \vec{PQ} = Q - P = (4 - 1) \hat{i} + (5 - 2) \hat{j} + (6 - 3) \hat{k} \]

Calculating the components:

\[ \vec{PQ} = 3 \hat{i} + 3 \hat{j} + 3 \hat{k} \]

### Step 2: Calculate the magnitude of \( \vec{PQ} \)

The 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 \( \vec{PQ} = 3 \hat{i} + 3 \hat{j} + 3 \hat{k} \):

\[ |\vec{PQ}| = \sqrt{3^2 + 3^2 + 3^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3} \]

### Step 3: Find the unit vector in the direction of \( \vec{PQ} \)

The unit vector \( \hat{u} \) in the direction of \( \vec{PQ} \) is given by:

\[ \hat{u} = \frac{\vec{PQ}}{|\vec{PQ}|} \]

Substituting the values we have:

\[ \hat{u} = \frac{3 \hat{i} + 3 \hat{j} + 3 \hat{k}}{3\sqrt{3}} = \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k} \]

### Final Answer

Thus, the unit vector in the direction of \( \vec{PQ} \) is:

\[ \hat{u} = \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k} \]

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