Question

Let two rods placed on the ground be represented by vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $-2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

Hint:

Always double-check that your cross product is orthogonal to the original vectors by taking the dot product. Here, $(-\hat{i} + 2\hat{j} + 2\hat{k}) \cdot (4\hat{i} - \hat{j} + 3\hat{k}) = -4 - 2 + 6 = 0$. This guarantees absolute algebraic accuracy!

Answer 1

Concept: To find a vector perpendicular to two given vectors $\vec{a}$ and $\vec{b}$, we use their cross product $\vec{a} \times \vec{b}$. The geometric significance of the cross product is that it produces a vector perpendicular to the plane containing $\vec{a}$ and $\vec{b}$.
• Cross Product Formula: $\vec{n} = \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
• Required Vector: A vector of a given magnitude $\lambda$ in the direction of the unit normal vector $\hat{n}$ is given by $\vec{V} = \pm \lambda \hat{n} = \pm \lambda \frac{\vec{n}}{|\vec{n}|}$.

Step 1: Compute the cross product of the two given vectors to find a perpendicular vector.
Let the first rod vector be $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$ and the second rod vector be $\vec{b} = -2\hat{i} + \hat{j} - 2\hat{k}$. Let $\vec{n}$ be the vector perpendicular to both $\vec{a}$ and $\vec{b}$: \[ \vec{n} = \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 4 & -1 & 3 -2 & 1 & -2 \end{vmatrix} \] Expanding this determinant along the first row: \[ \vec{n} = \hat{i} \left((-1)(-2) - (3)(1)\right) - \hat{j} \left((4)(-2) - (3)(-2)\right) + \hat{k} \left((4)(1) - (-1)(-2)\right) \] Simplifying the arithmetic inside the parentheses: \[ \vec{n} = \hat{i}(2 - 3) - \hat{j}(-8 + 6) + \hat{k}(4 - 2) \] \[ \vec{n} = -\hat{i} - \hat{j}(-2) + 2\hat{k} = -\hat{i} + 2\hat{j} + 2\hat{k} \]

Step 2: Calculate the magnitude of $\vec{n}$ and determine the final vector with a magnitude of 5.
Now, compute the magnitude of $\vec{n}$: \[ |\vec{n}| = \sqrt{(-1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \] The unit vector $\hat{n}$ pointing in the direction perpendicular to both rods is: \[ \hat{n} = \frac{\vec{n}}{|\vec{n}|} = \frac{-\hat{i} + 2\hat{j} + 2\hat{k}}{3} \] Since the flag-post must have a height of 5 m, the required vector representing the flag-post is: \[ \vec{V} = \pm 5 \hat{n} = \pm 5 \left( \frac{-\hat{i} + 2\hat{j} + 2\hat{k}}{3} \right) = \pm \frac{5}{3}(-\hat{i} + 2\hat{j} + 2\hat{k}) \]