Find a unit vector that is orthogonal to both i 1 j and i 1 k

To find a unit vector that is orthogonal to both vectors \( \mathbf{v_1} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v_2} = \mathbf{i} + \mathbf{k} \), we can follow these steps:

### Step 1: Find the cross product
The cross product of two vectors gives a vector that is orthogonal to both. Let's calculate the cross product \( \mathbf{v_1} \times \mathbf{v_2} \).

Given:
\[
\mathbf{v_1} = \mathbf{i} + \mathbf{j} = (1, 1, 0)
\]
\[
\mathbf{v_2} = \mathbf{i} + \mathbf{k} = (1, 0, 1)
\]

The cross product \( \mathbf{v_1} \times \mathbf{v_2} \) is given by the determinant of the following matrix:
\[
\mathbf{v_1} \times \mathbf{v_2} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 1 & 0 \\
1 & 0 & 1
\end{vmatrix}
\]

### Step 2: Calculate the determinant
Expanding the determinant, we get:
\[
\mathbf{v_1} \times \mathbf{v_2} = \mathbf{i} \left( 1 \times 1 - 0 \times 0 \right) - \mathbf{j} \left( 1 \times 1 - 0 \times 1 \right) + \mathbf{k} \left( 1 \times 0 - 1 \times 1 \right)
\]
\[
\mathbf{v_1} \times \mathbf{v_2} = \mathbf{i}(1) - \mathbf{j}(1) + \mathbf{k}(-1)
\]
\[
\mathbf{v_1} \times \mathbf{v_2} = \mathbf{i} - \mathbf{j} - \mathbf{k}
\]
Thus, the vector orthogonal to both \( \mathbf{v_1} \) and \( \mathbf{v_2} \) is \( \mathbf{i} - \mathbf{j} - \mathbf{k} \).

### Step 3: Find the unit vector
To find the unit vector, we need to divide this vector by its magnitude. The magnitude of \( \mathbf{i} - \mathbf{j} - \mathbf{k} \) is calculated as follows:
\[
\text{Magnitude} = \sqrt{(1)^2 + (-1)^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}
\]
Thus, the unit vector is:
\[
\frac{1}{\sqrt{3}} (\mathbf{i} - \mathbf{j} - \mathbf{k})
\]

### Final Answer:
The unit vector that is orthogonal to both \( \mathbf{i} + \mathbf{j} \) and \( \mathbf{i} + \mathbf{k} \) is:
\[
\frac{1}{\sqrt{3}} (\mathbf{i} - \mathbf{j} - \mathbf{k})
\]

Would you like any further clarification on this?

Here are some related questions to consider:
1. What would the unit vector be if the original vectors were different?
2. How does the cross product relate to the area of a parallelogram formed by two vectors?
3. How can you verify that the found unit vector is orthogonal to both original vectors?
4. What would happen if the two vectors were parallel?
5. How can you find a unit vector that is parallel to a given vector?

**Tip:** When finding a unit vector, always remember to normalize by dividing by the magnitude.

Learn how to find a unit vector that is orthogonal to both vectors i+j and i+k with a detailed step-by-step solution. This problem covers key concepts in vectors, including cross product calculation and unit vector normalization. Suitable for advanced high school students.

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