Question

The lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ are \dots}

• A. intersecting but not perpendicular
• B. perpendicular
• C. parallel
• D. skew lines

Hint:

To quickly test if lines intersect, equate the simplest coordinates first (like $y$ and $z$ here, which were trivial to solve) and use the remaining coordinate strictly as a "verification check" for the parameters.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two lines in 3D space in vector form. We need to determine their relative spatial relationship: whether they are parallel, perpendicular, skew, or intersecting.

Step 2: Key Formula or Approach:
1. Check for parallelism by examining if their direction vectors are scalar multiples.
2. Check for perpendicularity by calculating the dot product of their direction vectors (it should be 0).
3. Check for intersection by converting to parametric equations and solving for a common point.

Step 3: Detailed Explanation:
Let the direction vector of Line 1 be $\vec{d}_1 = 3\hat{i} - \hat{j}$.
Let the direction vector of Line 2 be $\vec{d}_2 = 2\hat{i} + 3\hat{k}$.
- They are clearly not scalar multiples of each other, so they are not parallel.
- Let's check the dot product: $\vec{d}_1 \cdot \vec{d}_2 = (3)(2) + (-1)(0) + (0)(3) = 6 \neq 0$. They are not perpendicular.
Now, let's check for intersection by writing the parametric equations $(x, y, z)$.
Line 1: $x = 1 + 3\lambda, \ y = 1 - \lambda, \ z = -1$
Line 2: $x = 4 + 2\mu, \ y = 0, \ z = -1 + 3\mu$
Equate the coordinates of both lines to see if a valid $(\lambda, \mu)$ pair exists:
Equating $y$-coordinates: $1 - \lambda = 0 \implies \lambda = 1$.
Equating $z$-coordinates: $-1 = -1 + 3\mu \implies 3\mu = 0 \implies \mu = 0$.
Now substitute $\lambda = 1$ and $\mu = 0$ into the $x$-coordinate equations to verify consistency:
Line 1 $x$-coord: $x = 1 + 3(1) = 4$.
Line 2 $x$-coord: $x = 4 + 2(0) = 4$.
Since the $x$-coordinates perfectly match, the lines physically intersect at the point $(4, 0, -1)$.

Step 4: Final Answer:
The lines intersect but are not perpendicular, matching option (a).