Line Definitions

The two lines are given by:

• $L_1$: $\vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$
• $L_2$: $\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + 2\hat{j} + \hat{k})$

In coordinate form:

• $L_1$: $x = 1 + 2\lambda, y = 2 + 3\lambda, z = 3 + 4\lambda$
• $L_2$: $x = 4 + 5\mu, y = 1 + 2\mu, z = \mu$

Intersection Point R Calculation

Equating the components of $L_1$ and $L_2$ to find the intersection point R:

1. $1 + 2\lambda = 4 + 5\mu \implies 2\lambda - 5\mu = 3$
2. $2 + 3\lambda = 1 + 2\mu \implies 3\lambda - 2\mu = -1$
3. $3 + 4\lambda = \mu$

Substitute equation (3) into equation (1):

$2\lambda - 5(3 + 4\lambda) = 3$

$2\lambda - 15 - 20\lambda = 3$

$-18\lambda = 18 \implies \lambda = -1$

Substitute $\lambda = -1$ into equation (3):

$\mu = 3 + 4(-1) = 3 - 4 = -1$

Substitute $\lambda = -1$ into the equation for $L_1$ to find R:

$x_R = 1 + 2(-1) = -1$

$y_R = 2 + 3(-1) = -1$

$z_R = 3 + 4(-1) = -1$

So, the intersection point is $R = (-1, -1, -1)$.

Point P on L1 Calculation

A general point P on $L_1$ is $P(\lambda) = (1 + 2\lambda, 2 + 3\lambda, 3 + 4\lambda)$.

The distance condition is $|\vec{PR}| = \sqrt{29}$.

$\vec{PR} = R - P(\lambda) = (-1 - (1 + 2\lambda), -1 - (2 + 3\lambda), -1 - (3 + 4\lambda))$

$\vec{PR} = (-2 - 2\lambda, -3 - 3\lambda, -4 - 4\lambda)$

$|\vec{PR}|^2 = (-2(1 + \lambda))^2 + (-3(1 + \lambda))^2 + (-4(1 + \lambda))^2 = 29$

$(4 + 9 + 16)(1 + \lambda)^2 = 29$

$29(1 + \lambda)^2 = 29 \implies (1 + \lambda)^2 = 1$

This gives $1 + \lambda = 1$ or $1 + \lambda = -1$.

So, $\lambda = 0$ or $\lambda = -2$.

The point P lies in the first octant (all coordinates positive).

• If $\lambda = 0$, $P = (1, 2, 3)$. This is in the first octant.
• If $\lambda = -2$, $P = (-3, -4, -5)$. Not in the first octant.

Therefore, $P = (1, 2, 3)$.

Point Q on L2 Calculation

A general point Q on $L_2$ is $Q(\mu) = (4 + 5\mu, 1 + 2\mu, \mu)$.

The distance condition is $|\vec{PQ}| = \sqrt{\frac{47}{3}}$.

$\vec{PQ} = Q(\mu) - P = (4 + 5\mu - 1, 1 + 2\mu - 2, \mu - 3)$

$\vec{PQ} = (3 + 5\mu, -1 + 2\mu, -3 + \mu)$

$|\vec{PQ}|^2 = (3 + 5\mu)^2 + (-1 + 2\mu)^2 + (-3 + \mu)^2 = \frac{47}{3}$

Expanding this:

$(9 + 30\mu + 25\mu^2) + (1 - 4\mu + 4\mu^2) + (9 - 6\mu + \mu^2) = \frac{47}{3}$

$30\mu^2 + 20\mu + 19 = \frac{47}{3}$

Multiply by 3:

$90\mu^2 + 60\mu + 57 = 47$

$90\mu^2 + 60\mu + 10 = 0$

Divide by 10:

$9\mu^2 + 6\mu + 1 = 0$

This is a perfect square: $(3\mu + 1)^2 = 0$.

Thus, $\mu = -\frac{1}{3}$.

The coordinates of Q are:

$x_Q = 4 + 5(-\frac{1}{3}) = 4 - \frac{5}{3} = \frac{7}{3}$

$y_Q = 1 + 2(-\frac{1}{3}) = 1 - \frac{2}{3} = \frac{1}{3}$

$z_Q = -\frac{1}{3}$

So, $Q = (\frac{7}{3}, \frac{1}{3}, -\frac{1}{3})$.

QR Squared Calculation

We have the coordinates of R and Q:

• $R = (-1, -1, -1)$
• $Q = (\frac{7}{3}, \frac{1}{3}, -\frac{1}{3})$

Calculate the vector $\vec{QR}$:

$\vec{QR} = R - Q = (-1 - \frac{7}{3}, -1 - \frac{1}{3}, -1 - (-\frac{1}{3}))$

$\vec{QR} = (-\frac{3+7}{3}, -\frac{3+1}{3}, -\frac{3-1}{3})$

$\vec{QR} = (-\frac{10}{3}, -\frac{4}{3}, -\frac{2}{3})$

Calculate the squared magnitude $|\vec{QR}|^2$:

$|\vec{QR}|^2 = (-\frac{10}{3})^2 + (-\frac{4}{3})^2 + (-\frac{2}{3})^2$

$|\vec{QR}|^2 = \frac{100}{9} + \frac{16}{9} + \frac{4}{9} = \frac{120}{9}$

Simplify the fraction:

$|\vec{QR}|^2 = \frac{40}{3}$

Final 27(QR)^2 Calculation

We need to find the value of $27(QR)^2$.

$27 \times |\vec{QR}|^2 = 27 \times \frac{40}{3}$

$27 \times \frac{40}{3} = 9 \times 40 = 360$