To find the number of unit vectors that are perpendicular to the given vectors \( \vec{a} = (1, 1, 0) \) and \( \vec{b} = (0, 1, 1) \), we will follow these steps:
### Step 1: Calculate the cross product of vectors \( \vec{a} \) and \( \vec{b} \).
The cross product \( \vec{a} \times \vec{b} \) is calculated using the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of vectors \( \vec{a} \) and \( \vec{b} \):
\[
\vec{a} \times \vec{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 0 \\
0 & 1 & 1
\end{vmatrix}
\]
Calculating this determinant, we have:
\[
\vec{a} \times \vec{b} = \hat{i} \begin{vmatrix}
1 & 0 \\
1 & 1
\end{vmatrix} - \hat{j} \begin{vmatrix}
1 & 0 \\
0 & 1
\end{vmatrix} + \hat{k} \begin{vmatrix}
1 & 1 \\
0 & 1
\end{vmatrix}
\]
Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} = (1)(1) - (0)(1) = 1 \)
2. \( \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = (1)(1) - (0)(0) = 1 \)
3. \( \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} = (1)(1) - (1)(0) = 1 \)
Putting it all together:
\[
\vec{a} \times \vec{b} = \hat{i}(1) - \hat{j}(1) + \hat{k}(1) = (1, -1, 1)
\]
### Step 2: Find the magnitude of the cross product.
The magnitude of \( \vec{a} \times \vec{b} \) is given by:
\[
|\vec{a} \times \vec{b}| = \sqrt{(1)^2 + (-1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}
\]
### Step 3: Determine the unit vector in the direction of \( \vec{a} \times \vec{b} \).
To find the unit vector \( \hat{n} \) in the direction of \( \vec{a} \times \vec{b} \):
\[
\hat{n} = \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} = \frac{(1, -1, 1)}{\sqrt{3}} = \left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)
\]
### Step 4: Identify the vectors perpendicular to both \( \vec{a} \) and \( \vec{b} \).
Since unit vectors can point in both directions, there are two unit vectors perpendicular to both \( \vec{a} \) and \( \vec{b} \):
1. \( \hat{n} = \left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \)
2. \( -\hat{n} = \left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right) \)
### Conclusion
Thus, the number of unit vectors of unit length that are perpendicular to both \( \vec{a} \) and \( \vec{b} \) is **2**.
### Answer: b. two
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