© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Write the Parametric Forms of the Given Lines
The first line, L_1 , is given by
\overrightarrow{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}).
In coordinate form, a general point on L_1 can be written as
\bigl(1 + \lambda,\; 2 - \lambda,\; 3 + \lambda\bigr).
The second line, L_2 , is
\overrightarrow{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k}).
A general point on L_2 can be written as
\bigl(4 + \mu,\; 5 + \mu,\; 6 - \mu\bigr).
Step 2: Represent Points P and Q on the Respective Lines
Let P be the point on L_1 and Q the point on L_2 such that the line segment PQ is the common perpendicular (i.e., the shortest distance) between L_1 and L_2 . Thus,
P = \bigl(1 + \lambda,\; 2 - \lambda,\; 3 + \lambda\bigr),
\quad
Q = \bigl(4 + \mu,\; 5 + \mu,\; 6 - \mu\bigr).
Step 3: Use the Perpendicularity Condition
The vector
\overrightarrow{PQ}
= \bigl((4 + \mu) - (1 + \lambda),\; (5 + \mu) - (2 - \lambda),\; (6 - \mu) - (3 + \lambda)\bigr)
must be perpendicular to the direction vectors of both lines.
Direction of L_1 : (1,\,-1,\,1)
Direction of L_2 : (1,\;1,\,-1)
Hence,
\overrightarrow{PQ}\cdot(1,\,-1,\,1) = 0,
\quad
\overrightarrow{PQ}\cdot(1,\;1,\,-1) = 0.
Step 4: Form and Solve the System of Equations
First, simplify \overrightarrow{PQ} :
\[
\overrightarrow{PQ} = \bigl((4 + \mu) - (1 + \lambda),\; (5 + \mu) - (2 - \lambda),\; (6 - \mu) - (3 + \lambda)\bigr)
= (3 + \mu - \lambda,\; 3 + \mu + \lambda,\; 3 - \mu - \lambda).
\]
1) Perpendicular to (1, -1, 1) :
\[
(3 + \mu - \lambda)\times 1 \;+\; (3 + \mu + \lambda)\times(-1) \;+\; (3 - \mu - \lambda)\times 1 = 0.
\]
Which simplifies to
\[
3 + \mu - \lambda - (3 + \mu + \lambda) + (3 - \mu - \lambda) = 0
\;\;\Longrightarrow\;\;
3 - \mu - 3\lambda = 0.
\]
2) Perpendicular to (1, 1, -1) :
\[
(3 + \mu - \lambda)\times 1 + (3 + \mu + \lambda)\times 1 + (3 - \mu - \lambda)\times(-1) = 0.
\]
Which simplifies to
\[
(3 + \mu - \lambda) + (3 + \mu + \lambda) - (3 - \mu - \lambda) = 0
\;\;\Longrightarrow\;\;
3 + 3\mu + \lambda = 0.
\]
Hence, the system is:
\[
3 - \mu - 3\lambda = 0,
\quad
3 + 3\mu + \lambda = 0.
\]
From the first equation,
\[
\mu = 3 - 3\lambda.
\]
Substitute this into the second:
\[
3 + 3(3 - 3\lambda) + \lambda = 0
\;\Longrightarrow\;
3 + 9 - 9\lambda + \lambda = 0
\;\Longrightarrow\;
12 - 8\lambda = 0
\;\Longrightarrow\;
\lambda = \frac{3}{2}.
\]
Then
\[
\mu = 3 - 3\left(\frac{3}{2}\right) = 3 - \frac{9}{2} = -\frac{3}{2}.
\]
Step 5: Find Coordinates of P and Q
Substitute \lambda = \tfrac{3}{2} into the expression for P :
\[
P = \Bigl(1 + \tfrac{3}{2},\; 2 - \tfrac{3}{2},\; 3 + \tfrac{3}{2}\Bigr)
= \bigl(2.5,\; 0.5,\; 4.5 \bigr).
\]
Substitute \mu = -\tfrac{3}{2} into the expression for Q :
\[
Q = \Bigl(4 + \bigl(-\tfrac{3}{2}\bigr),\; 5 + \bigl(-\tfrac{3}{2}\bigr),\; 6 - \bigl(-\tfrac{3}{2}\bigr)\Bigr)
= \bigl(2.5,\; 3.5,\; 7.5 \bigr).
\]
Step 6: Determine the Midpoint and Compute the Required Quantity
Let the midpoint of segment PQ be M(\alpha, \beta, \gamma) . Then
\[
M = \Bigl(\frac{2.5 + 2.5}{2},\; \frac{0.5 + 3.5}{2},\; \frac{4.5 + 7.5}{2}\Bigr)
= (2.5,\; 2,\; 6).
\]
Therefore, \alpha = 2.5,\; \beta = 2,\; \gamma = 6.
So,
\[
\alpha + \beta + \gamma = 2.5 + 2 + 6 = 10.5.
\]
Finally,
\[
2(\alpha + \beta + \gamma) = 2 \times 10.5 = 21.
\]
Final Answer
2(\alpha + \beta + \gamma) = 21.
