The magnitude of resultant vectors of two vectors given by `vecA = 10hati+15hatj` and `vecB = 5hati` would be

Options

A. 15
B. `15sqrt2`
C. `12sqrt2`
D. 10

Answer

To find the magnitude of the resultant vector of the two vectors \(\vec{A} = 10\hat{i} + 15\hat{j}\) and \(\vec{B} = 5\hat{i}\), we can follow these steps:

### Step 1: Add the Vectors
First, we need to add the two vectors \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = \vec{A} + \vec{B} = (10\hat{i} + 15\hat{j}) + (5\hat{i}) \]
Combine the \(\hat{i}\) components and the \(\hat{j}\) components:
\[ \vec{R} = (10 + 5)\hat{i} + 15\hat{j} = 15\hat{i} + 15\hat{j} \]

### Step 2: Write the Resultant Vector
Now we have the resultant vector:
\[ \vec{R} = 15\hat{i} + 15\hat{j} \]

### Step 3: Find the Magnitude of the Resultant Vector
To find the magnitude of the resultant vector \(\vec{R}\), we use the formula for the magnitude of a vector:
\[ |\vec{R}| = \sqrt{(R_x)^2 + (R_y)^2} \]
where \(R_x\) and \(R_y\) are the components of the vector. Here, \(R_x = 15\) and \(R_y = 15\):
\[ |\vec{R}| = \sqrt{(15)^2 + (15)^2} = \sqrt{225 + 225} = \sqrt{450} \]

### Step 4: Simplify the Magnitude
Now, simplify \(\sqrt{450}\):
\[ \sqrt{450} = \sqrt{225 \times 2} = \sqrt{225} \times \sqrt{2} = 15\sqrt{2} \]

### Final Answer
Thus, the magnitude of the resultant vector is:
\[ |\vec{R}| = 15\sqrt{2} \]

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