Question

The shortest distance of the point P(0,1,2) from the line L is equal to -

• A. $\frac{\sqrt{119}}{3}$
• B. 2
• C. $\frac{\sqrt{219}}{3}$
• D. $\frac{\sqrt{13}}{3}$

Answer 1

Answer: (C)

$\frac{\sqrt{219}}{3}$

Answer 2

The equation of the line L is given by $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$. This can be rewritten as $(\vec{r} - \vec{a}) \times \vec{b} = \vec{0}$, implying that $(\vec{r} - \vec{a})$ is parallel to $\vec{b}$. Thus, the equation of the line L is $\vec{r} = \vec{a} + t\vec{b}$.

Given $\vec{a} = \hat{i} +2\hat{j} -3\hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$, the line L passes through the point A(1, 2, -3) and is parallel to $\vec{b} = (2, -1, 1)$.

The shortest distance $d$ of a point P with position vector $\vec{p}$ from a line passing through point A with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is given by:

$d = \frac{|(\vec{p} - \vec{a}) \times \vec{b}|}{|\vec{b}|}$

1. Calculate $\vec{p} - \vec{a} = (0-1)\hat{i} + (1-2)\hat{j} + (2-(-3))\hat{k} = -\hat{i} - \hat{j} + 5\hat{k}$.

2. Calculate $(\vec{p} - \vec{a}) \times \vec{b} = (-\hat{i} - \hat{j} + 5\hat{k}) \times (2\hat{i} - \hat{j} + \hat{k}) = 4\hat{i} + 11\hat{j} + 3\hat{k}$.

3. Calculate $|(\vec{p} - \vec{a}) \times \vec{b}| = \sqrt{4^2 + 11^2 + 3^2} = \sqrt{146}$.

4. Calculate $|\vec{b}| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{6}$.

5. Compute the distance $d = \frac{\sqrt{146}}{\sqrt{6}} = \sqrt{\frac{146}{6}} = \sqrt{\frac{73}{3}} = \frac{\sqrt{219}}{3}$.

Therefore, the shortest distance is $\frac{\sqrt{219}}{3}$.