given to vector A=2i+3j and B=i-2j obtain a unit vector perpendicular to the to vector

To find a **unit vector perpendicular** to both vectors \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{B} = \mathbf{i} - 2\mathbf{j} \), let us proceed step-by-step:

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### Step 1: Find the Cross Product of \( \mathbf{A} \) and \( \mathbf{B} \)
Since both vectors \( \mathbf{A} \) and \( \mathbf{B} \) lie in the **2D plane**, the cross product will result in a vector along the **z-axis**. In 2D, we compute the magnitude of the cross product as:

\[
|\mathbf{A} \times \mathbf{B}| = (2)(-2) - (3)(1) = -4 - 3 = -7
\]

This means that the cross product vector (along the z-axis) is:

\[
\mathbf{A} \times \mathbf{B} = -7 \mathbf{k}
\]

Thus, the perpendicular vector to both \( \mathbf{A} \) and \( \mathbf{B} \) is along the **negative z-direction**: \( -7\mathbf{k} \).

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### Step 2: Convert to a Unit Vector
A unit vector is obtained by dividing the vector by its magnitude.

\[
\text{Magnitude of } -7\mathbf{k} = | -7 | = 7
\]

Thus, the unit vector along \( -7\mathbf{k} \) is:

\[
\hat{\mathbf{n}} = \frac{-7 \mathbf{k}}{7} = -\mathbf{k}
\]

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### Final Answer
The **unit vector perpendicular** to both \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{B} = \mathbf{i} - 2\mathbf{j} \) is:

\[
\boxed{-\mathbf{k}}
\]

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Would you like further explanation on how the cross product works or more practice on unit vectors?  

Here are five related questions you might find useful:  
1. How do you find the magnitude of a vector?  
2. What does it mean for two vectors to be orthogonal?  
3. How do you find the dot product of two vectors?  
4. What is the difference between a cross product and a dot product?  
5. How do you find the projection of one vector onto another?  

**Tip:** The cross product is only defined in **3D** space, but the result in 2D scenarios is typically treated as the magnitude in the z-direction!

Learn how to find a unit vector perpendicular to the given vectors A = 2i + 3j and B = i - 2j using cross product and unit vector concepts. This solution includes detailed steps and is suitable for students in Grades 10-12 studying vector operations.

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