r=(i+j-k)+ (ai-j),a 0 and r=(4i-k)+ (2i+ak) from the origin is :

Question

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (a\hat{i}-\hat{j}),a
e 0$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+a\hat{k})$ from the origin is :

Accepted Answer

For the first line, the equation is:

${\vec{r}}_{1}=(\hat{i}+\hat{j}-\hat{k})+\lambda (a\hat{i}-\hat{j})$

We can group the components along the $\hat{i}$, $\hat{j}$, and $\hat{k}$ directions:

${\vec{r}}_{1}=(1+a\lambda )\hat{i}+(1-\lambda )\hat{j}-\hat{k}$

For the second line, the equation is:

${\vec{r}}_{2}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+a\hat{k})$

Grouping the components similarly:

${\vec{r}}_{2}=(4+2\mu )\hat{i}+0\hat{j}+(a\mu -1)\hat{k}$

If the two lines intersect, there must be a common point. This means that for some value of $\lambda $ and $\mu $, their position vectors will be exactly the same. We can find this by equating the corresponding components of ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$:

Equating the $\hat{j}$ components:

$1-\lambda =0$

$\lambda =1$

Equating the $\hat{k}$ components:

$-1=a\mu -1$

$a\mu =0$

Since the question specifies that $a
e 0$, it must be true that $\mu =0$.

Equating the $\hat{i}$ components:

$1+a\lambda =4+2\mu $

Now, substitute the values of $\lambda =1$ and $\mu =0$ into this equation:

$1+a(1)=4+2(0)$

$1+a=4$

$a=3$

Now that we have $\mu =0$ (and $\lambda =1$), we can substitute either of these back into the respective line equations to find the position vector of the point of intersection. Let us use the second line's equation by substituting $\mu =0$:

$\vec{r}=(4+2(0))\hat{i}+(3(0)-1)\hat{k}$

$\vec{r}=4\hat{i}-\hat{k}$

So, the coordinates of the point of intersection are $(4,0,-1)$.

Next, we calculate the distance $d$ of this point of intersection from the origin $(0,0,0)$ using the distance formula:

$d=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

$d=\sqrt{(4{)}^{2}+(0{)}^{2}+(-1{)}^{2}}$

$d=\sqrt{16+0+1}$

$d=\sqrt{17}$

Looking at the given options, the problem asks for the square of the distance from the origin.

The square of the distance is:

${d}^{2}=(\sqrt{17}{)}^{2}=17$

Therefore, the correct choice is Option C.

Solutions · 1