Unit vector perpendicular to $2\hat{i}-\hat{j}+\hat{k}$ and $3\hat{i}+4\hat{j}-\hat{k}$ is:

Question

Accepted Answer

To determine the unit vector perpendicular to the vectors $ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} $ and $ \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} $, we need to find the cross product of these two vectors. The cross product will give us a vector that is perpendicular to both $ \mathbf{a} $ and $ \mathbf{b} $.

The cross product $ \mathbf{a} \times \mathbf{b} $ is calculated as follows:

$$
\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \
2 & -1 & 1 \
3 & 4 & -1
\end{vmatrix}
$$

Calculating the determinant, we have:

$$
\mathbf{a} \times \mathbf{b} = \hat{i}((-1)(-1) - (1)(4)) - \hat{j}((2)(-1) - (1)(3)) + \hat{k}((2)(4) - (-1)(3))
$$

$$
= \hat{i}(1 - 4) - \hat{j}(-2 - 3) + \hat{k}(8 + 3)
$$

$$
= \hat{i}(-3) + \hat{j}(5) + \hat{k}(11)
$$

$$
= -3\hat{i} + 5\hat{j} + 11\hat{k}
$$

The magnitude of this vector is:

$$
\sqrt{(-3)^2 + 5^2 + 11^2} = \sqrt{9 + 25 + 121} = \sqrt{155}
$$

Thus, the unit vector perpendicular to both $ \mathbf{a} $ and $ \mathbf{b} $ is:

$$
\frac{-3\hat{i} + 5\hat{j} + 11\hat{k}}{\sqrt{155}}
$$

This matches Option 3, confirming it as the correct answer.

**Explanation for Incorrect Options:**

1. **Option 1: $\frac{2\hat{i}-\hat{j}+3\hat{k}}{\sqrt{15}}$**

This vector does not result from the cross product of $ \mathbf{a} $ and $ \mathbf{b} $. The components do not match the calculated perpendicular vector.

2. **Option 2: $\frac{-\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$**

This vector also does not match the result of the cross product. The components are incorrect, and the magnitude does not correspond to the calculated magnitude of the perpendicular vector.

3. **Option 4: $\frac{-3\hat{i}+4\hat{j}-3\hat{k}}{\sqrt{34}}$**

While the $ \hat{i} $ component is similar, the $ \hat{j} $ and $ \hat{k} $ components do not match the calculated vector. The magnitude is also incorrect.

In conclusion, Option 3 is the only choice that correctly represents the unit vector perpendicular to both given vectors, as derived from their cross product.