11. Given : j ˆ i 3 ˆ A  2   and j ˆ i 6 ˆ B  5   . The magnitude of A B    is

To find the magnitude of the vector sum \( \mathbf{A} + \mathbf{B} \), we first need to break down the given vectors \( \mathbf{A} \) and \( \mathbf{B} \) into their components. The vectors are expressed in terms of unit vectors \( \hat{i} \) and \( \hat{j} \), which represent the x and y directions, respectively.

Understanding the Vectors

The vectors provided are:

• \( \mathbf{A} = 2\hat{i} + 3\hat{j} \)
• \( \mathbf{B} = 5\hat{i} - 6\hat{j} \)

Here, \( \hat{i} \) corresponds to the x-component and \( \hat{j} \) corresponds to the y-component. Now, let’s find the components of each vector.

Finding the Components

For vector \( \mathbf{A} \):

• x-component: 2
• y-component: 3

For vector \( \mathbf{B} \):

• x-component: 5
• y-component: -6

Calculating the Sum of the Vectors

Next, we add the corresponding components of \( \mathbf{A} \) and \( \mathbf{B} \) to find \( \mathbf{A} + \mathbf{B} \):

• x-component: \( 2 + 5 = 7 \)
• y-component: \( 3 - 6 = -3 \)

Thus, the resultant vector \( \mathbf{A} + \mathbf{B} \) can be expressed as:

\( \mathbf{A} + \mathbf{B} = 7\hat{i} - 3\hat{j} \)

Finding the Magnitude

The magnitude of a vector \( \mathbf{V} = x\hat{i} + y\hat{j} \) is calculated using the formula:

\( |\mathbf{V}| = \sqrt{x^2 + y^2} \)

Applying this to our resultant vector \( \mathbf{A} + \mathbf{B} \):

• x-component: 7
• y-component: -3

Now, substituting these values into the magnitude formula:

\( |\mathbf{A} + \mathbf{B}| = \sqrt{7^2 + (-3)^2} \)

Calculating this gives:

• \( 7^2 = 49 \)
• \( (-3)^2 = 9 \)

So, we have:

\( |\mathbf{A} + \mathbf{B}| = \sqrt{49 + 9} = \sqrt{58} \)

Final Result

The magnitude of the vector sum \( \mathbf{A} + \mathbf{B} \) is \( \sqrt{58} \), which is approximately 7.62 when calculated numerically. This value represents the length of the resultant vector in the Cartesian plane.