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 from the two given vectors \(\vec{A} = 10\hat{i} + 15\hat{j}\) and \(\vec{B} = 5\hat{i}\), we can follow these steps:

### Step 1: Write down the vectors
We have:
\[ \vec{A} = 10\hat{i} + 15\hat{j} \]
\[ \vec{B} = 5\hat{i} \]

### Step 2: Add the vectors component-wise
To find the resultant vector \(\vec{R} = \vec{A} + \vec{B}\), we add the components in the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) directions.

- For the \(\hat{i}\) component:
\[ R_x = A_x + B_x = 10 + 5 = 15 \]

- For the \(\hat{j}\) component:
\[ R_y = A_y + B_y = 15 + 0 = 15 \]

- For the \(\hat{k}\) component (which is zero for both vectors):
\[ R_z = A_z + B_z = 0 + 0 = 0 \]

Thus, the resultant vector \(\vec{R}\) is:
\[ \vec{R} = 15\hat{i} + 15\hat{j} + 0\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
The magnitude of a vector \(\vec{R} = R_x\hat{i} + R_y\hat{j} + R_z\hat{k}\) is given by the formula:
\[ |\vec{R}| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]

Substituting the values we found:
\[ |\vec{R}| = \sqrt{(15)^2 + (15)^2 + (0)^2} \]
\[ |\vec{R}| = \sqrt{225 + 225 + 0} = \sqrt{450} \]
\[ |\vec{R}| = 15\sqrt{2} \]

### Final Answer
The magnitude of the resultant vector is \(15\sqrt{2}\).

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