Two vectors are given by `vec(A) = hat(i) + 2hat(j) + 2hat(k)` and `vec(B) = 3hat(i) + 6hat(j) + 2hat(k)`. Another vector `vec(C )` has the same magnitude as `vec(B)` but has the same direction as `vec(A)`. Then which of the following vectors represents `vec(C )` ?

Options

A. `(7)/(3)(hat(i) + 2hat(j) + 2hat(k))`
B. `(3)/(7)(hat(i) - 2hat(j) + 2hat(k))`
C. `(7)/(9)(hat(i) - 2hat(j) + 2hat(k))`
D. `(7)/(9)(hat(i) + 2hat(j) + 2hat(k))`

Answer

To find the vector \(\vec{C}\) that has the same magnitude as \(\vec{B}\) but the same direction as \(\vec{A}\), we can follow these steps:

### Step 1: Calculate the magnitude of vector \(\vec{B}\)

The magnitude of a vector \(\vec{B} = a\hat{i} + b\hat{j} + c\hat{k}\) is given by the formula:

\[ |\vec{B}| = \sqrt{a^2 + b^2 + c^2} \]

For \(\vec{B} = 3\hat{i} + 6\hat{j} + 2\hat{k}\):

\[ |\vec{B}| = \sqrt{3^2 + 6^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

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

For \(\vec{A} = \hat{i} + 2\hat{j} + 2\hat{k}\):

\[ |\vec{A}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

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

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

\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3} = \frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k} \]

### Step 4: Scale the unit vector \(\hat{A}\) to have the same magnitude as \(\vec{B}\)

To find \(\vec{C}\), we scale the unit vector \(\hat{A}\) by the magnitude of \(\vec{B}\):

\[ \vec{C} = |\vec{B}| \cdot \hat{A} = 7 \cdot \left(\frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\right) \]

Calculating this gives:

\[ \vec{C} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

### Final Answer

Thus, the vector \(\vec{C}\) is:

\[ \vec{C} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

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