The component of `vec(A)=hati+hatj+5hatk` perpendicular to `vec(B)=3hati+4hatj` is

Options

A. `-(4)/(25)hati+(3)/(25)hatj+5hatk`
B. `-(8)/(25)hati-(6)/(25)hatj+5hatk`
C. `(4)/(25)hati-(3)/(25)hatj+5hatk`
D. `+(8)/(25)hati-(6)/(25)hatj+5hatk`

Explanation

To find the component of vector \( \vec{A} = \hat{i} + \hat{j} + 5\hat{k} \) that is perpendicular to vector \( \vec{B} = 3\hat{i} + 4\hat{j} \), we can follow these steps:

### Step 1: Calculate the dot product of \( \vec{A} \) and \( \vec{B} \)

The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows:

\[ \vec{A} \cdot \vec{B} = (1)(3) + (1)(4) + (5)(0) = 3 + 4 + 0 = 7 \]

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

The magnitude of \( \vec{B} \) is given by:

\[ |\vec{B}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate the parallel component of \( \vec{A} \) along \( \vec{B} \)

The formula for the magnitude of the parallel component of \( \vec{A} \) along \( \vec{B} \) is:

\[ |\vec{A}_{\parallel}| = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{7}{5} \]

### Step 4: Find the unit vector in the direction of \( \vec{B} \)

The unit vector \( \hat{b} \) in the direction of \( \vec{B} \) is:

\[ \hat{b} = \frac{\vec{B}}{|\vec{B}|} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \]

### Step 5: Calculate the parallel component vector of \( \vec{A} \)

Now we can find the parallel component vector \( \vec{A}_{\parallel} \):

\[ \vec{A}_{\parallel} = |\vec{A}_{\parallel}| \hat{b} = \frac{7}{5} \left( \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \right) = \frac{21}{25}\hat{i} + \frac{28}{25}\hat{j} \]

### Step 6: Calculate the perpendicular component of \( \vec{A} \)

The perpendicular component \( \vec{A}_{\perpendicular} \) can be found using the formula:

\[ \vec{A}_{\perpendicular} = \vec{A} - \vec{A}_{\parallel} \]

Substituting the values we have:

\[ \vec{A}_{\perpendicular} = \left( \hat{i} + \hat{j} + 5\hat{k} \right) - \left( \frac{21}{25}\hat{i} + \frac{28}{25}\hat{j} \right) \]

### Step 7: Simplify the expression

Now we simplify the expression:

\[ \vec{A}_{\perpendicular} = \left( 1 - \frac{21}{25} \right) \hat{i} + \left( 1 - \frac{28}{25} \right) \hat{j} + 5\hat{k} \]

Calculating each component:

\[ 1 - \frac{21}{25} = \frac{25 - 21}{25} = \frac{4}{25} \]
\[ 1 - \frac{28}{25} = \frac{25 - 28}{25} = -\frac{3}{25} \]

Thus, we have:

\[ \vec{A}_{\perpendicular} = \frac{4}{25}\hat{i} - \frac{3}{25}\hat{j} + 5\hat{k} \]

### Final Answer

The component of \( \vec{A} \) that is perpendicular to \( \vec{B} \) is:

\[ \vec{A}_{\perpendicular} = \frac{4}{25}\hat{i} - \frac{3}{25}\hat{j} + 5\hat{k} \]

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