For any particle moving with some velocity `(vecv)` & acceleration `(veca)`, tangential acceleration & normal acceleration are defined as follows.
Tangential acceleration - The component of acceleration in the direction of velocity.
Normal acceleration - The component of acceleration in the direction perpendicular to velocity.
If at a given instant, velocity & acceleration of a particle are given by .
`vecc=4hati +3hatj`
`veca=10hati+15hatj+20hatk`
Find the normal acceleration of the particles at the given instant :-

Options

A. `(-9hati+12hatj+50hatk)/(5)`
B. `(-9hati-12hatj-50hatk)/(5)`
C. `(-18hati+24hatj+100hatk)/(5)`
D. `(18hati-24hatj-100hatk)/(5)`

Answer

To find the normal acceleration of a particle given its velocity and acceleration vectors, we can follow these steps:

### Step 1: Identify the given vectors
The velocity vector \( \vec{v} \) and acceleration vector \( \vec{a} \) are given as:
\[ \vec{v} = 4 \hat{i} + 3 \hat{j} \]
\[ \vec{a} = 10 \hat{i} + 15 \hat{j} + 20 \hat{k} \]

### Step 2: Calculate the magnitude of the velocity vector
The magnitude of the velocity vector \( |\vec{v}| \) is calculated as:
\[ |\vec{v}| = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Calculate the dot product of acceleration and velocity vectors
The dot product \( \vec{a} \cdot \vec{v} \) is calculated as follows:
\[ \vec{a} \cdot \vec{v} = (10 \hat{i} + 15 \hat{j} + 20 \hat{k}) \cdot (4 \hat{i} + 3 \hat{j}) = 10 \cdot 4 + 15 \cdot 3 + 20 \cdot 0 = 40 + 45 + 0 = 85 \]

### Step 4: Calculate the tangential acceleration
The tangential acceleration \( a_t \) can be found using the formula:
\[ a_t = \frac{\vec{a} \cdot \vec{v}}{|\vec{v}|} \]
Substituting the values:
\[ a_t = \frac{85}{5} = 17 \]

### Step 5: Find the tangential acceleration vector
The tangential acceleration vector \( \vec{a_t} \) is in the direction of the velocity vector:
\[ \vec{a_t} = a_t \cdot \frac{\vec{v}}{|\vec{v}|} = 17 \cdot \frac{(4 \hat{i} + 3 \hat{j})}{5} = \frac{68}{5} \hat{i} + \frac{51}{5} \hat{j} \]

### Step 6: Calculate the normal acceleration
The normal acceleration \( \vec{a_n} \) can be found using the relationship:
\[ \vec{a} = \vec{a_t} + \vec{a_n} \]
Thus,
\[ \vec{a_n} = \vec{a} - \vec{a_t} \]
Substituting the values:
\[ \vec{a_n} = (10 \hat{i} + 15 \hat{j} + 20 \hat{k}) - \left(\frac{68}{5} \hat{i} + \frac{51}{5} \hat{j}\right) \]
Calculating each component:
1. For the \( \hat{i} \) component:
\[ 10 - \frac{68}{5} = \frac{50}{5} - \frac{68}{5} = -\frac{18}{5} \]
2. For the \( \hat{j} \) component:
\[ 15 - \frac{51}{5} = \frac{75}{5} - \frac{51}{5} = \frac{24}{5} \]
3. For the \( \hat{k} \) component:
\[ 20 - 0 = 20 = \frac{100}{5} \]

Thus, we have:
\[ \vec{a_n} = -\frac{18}{5} \hat{i} + \frac{24}{5} \hat{j} + 20 \hat{k} \]

### Final Answer
The normal acceleration vector is:
\[ \vec{a_n} = -\frac{18}{5} \hat{i} + \frac{24}{5} \hat{j} + 20 \hat{k} \]

---