Question

What is the vector perpendicular to both the vectors $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k}$?

• A. $-10\hat{i} - 3\hat{j} + 4\hat{k}$
• B. $-10\hat{i} + 3\hat{j} + 4\hat{k}$
• C. $10\hat{i} - 3\hat{j} + 4\hat{k}$
• D. None of the above

Answer 1

Answer: (A)

$-10\hat{i} - 3\hat{j} + 4\hat{k}$

Answer 2

To find a vector perpendicular to both $\vec{a}$ and $\vec{b}$, we compute their cross product $\vec{a} \times \vec{b}$.

Given vectors: $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ $\vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k}$

The cross product $\vec{a} \times \vec{b}$ is calculated as the determinant:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 4 & -4 & 7 \end{vmatrix}$$

Expanding the determinant:

$$\vec{a} \times \vec{b} = \hat{i}((-2)(7) - (1)(-4)) - \hat{j}((1)(7) - (1)(4)) + \hat{k}((1)(-4) - (-2)(4))$$ $$\vec{a} \times \vec{b} = \hat{i}(-14 + 4) - \hat{j}(7 - 4) + \hat{k}(-4 + 8)$$ $$\vec{a} \times \vec{b} = -10\hat{i} - 3\hat{j} + 4\hat{k}$$

Thus, the vector perpendicular to both $\vec{a}$ and $\vec{b}$ is $-10\hat{i} - 3\hat{j} + 4\hat{k}$.