If `vec(c )` is the unit vector perpendicular to both the vector `vec(a) and vec(b)`, then what is another unit vector perpendicular to both the vectors `vec(a) and vec(b)` ?

Options

A. `vec(c ) xx vec(a)`
B. `vec(c ) xx vec(d)`
C. `-((vec(a) xx vec(b)))/(|vec(a) xx vec(b)|)`
D. `((vec(a) xx vec(b)))/((vec(a) xx vec(b)))`

Answer

To find another unit vector perpendicular to both vectors \(\vec{a}\) and \(\vec{b}\), we can use the properties of the cross product. Here’s a step-by-step solution:

### Step 1: Understand the Cross Product
The cross product of two vectors \(\vec{a}\) and \(\vec{b}\) gives a vector that is perpendicular to both \(\vec{a}\) and \(\vec{b}\). This can be expressed as:
\[ \vec{c} = \vec{a} \times \vec{b} \]
where \(\vec{c}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\).

**Hint:** Remember that the cross product of two vectors results in a vector that is orthogonal to both.

### Step 2: Determine the Unit Vector
A unit vector in the direction of \(\vec{c}\) is given by:
\[ \hat{c} = \frac{\vec{c}}{|\vec{c}|} = \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} \]
This vector \(\hat{c}\) is the unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\).

**Hint:** To find a unit vector, divide the vector by its magnitude.

### Step 3: Find Another Unit Vector
The cross product has a property that allows us to find another vector that is also perpendicular to both \(\vec{a}\) and \(\vec{b}\). Specifically, we can take the negative of the unit vector:
\[ -\hat{c} = -\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} \]
This vector is also a unit vector and is perpendicular to both \(\vec{a}\) and \(\vec{b}\).

**Hint:** The negative of a vector points in the opposite direction but retains the same magnitude.

### Conclusion
Thus, another unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is:
\[ -\hat{c} = -\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} \]

### Final Answer
The answer is:
\[ -\hat{c} \]