The length of vector, `vecA= 3hati +4hatj + 9hatk` in y-z plane is

Options

A. `sqrt30` units
B. `sqrt12 `units
C. `sqrt97` units
D. `sqrt50 `units

Answer

To find the length of the vector \(\vec{A} = 3\hat{i} + 4\hat{j} + 9\hat{k}\) in the y-z plane, we can follow these steps:

### Step 1: Understand the Components of the Vector
The vector \(\vec{A}\) has three components:
- \(3\hat{i}\) in the x-direction
- \(4\hat{j}\) in the y-direction
- \(9\hat{k}\) in the z-direction

### Step 2: Identify the Relevant Plane
In the y-z plane, the x-component is not considered. Therefore, we only focus on the y and z components of the vector.

### Step 3: Write the Vector in the y-z Plane
In the y-z plane, the vector can be represented as:
\[ \vec{A}_{yz} = 4\hat{j} + 9\hat{k} \]

### Step 4: Calculate the Length of the Vector in the y-z Plane
The length (magnitude) of a vector \(\vec{V} = a\hat{j} + b\hat{k}\) can be calculated using the formula:
\[ |\vec{V}| = \sqrt{a^2 + b^2} \]
For our vector \(\vec{A}_{yz}\):
- \(a = 4\)
- \(b = 9\)

Now substituting these values into the formula:
\[ |\vec{A}_{yz}| = \sqrt{(4)^2 + (9)^2} = \sqrt{16 + 81} = \sqrt{97} \]

### Step 5: Conclusion
Thus, the length of the vector \(\vec{A}\) in the y-z plane is:
\[ \sqrt{97} \]