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CBSE Sample Paper 2023-24 · Section A · 1 mark

Q5: Lines with directions 2i^+3j^−6k^2\hat i + 3\hat j - 6\hat k and 6i^+9j^−18k^6\hat i + 9\hat j - 18\hat k

Sample 2023-24Ch 11 · Three Dimensional Geometry
The question

The lines r⃗=i^+j^−k^+λ(2i^+3j^−6k^)\vec r = \hat i + \hat j - \hat k + \lambda\left(2\hat i + 3\hat j - 6\hat k\right) and r⃗=2i^−j^−k^+μ(6i^+9j^−18k^)\vec r = 2\hat i - \hat j - \hat k + \mu\left(6\hat i + 9\hat j - 18\hat k\right)⁠; (where λ\lambda & μ\mu are scalars) are

(a) coincident

(b) skew

(c) intersecting

(d) parallel

Exam answer

The direction vectors are b⃗1=2i^+3j^−6k^\vec b_1 = 2\hat i + 3\hat j - 6\hat k and b⃗2=6i^+9j^−18k^\vec b_2 = 6\hat i + 9\hat j - 18\hat k⁠.

6i^+9j^−18k^=3(2i^+3j^−6k^)\begin{aligned} &6\hat i + 9\hat j - 18\hat k \\ &= 3\left(2\hat i + 3\hat j - 6\hat k\right) \end{aligned}
Why?

Two lines are parallel when their direction vectors are scalar multiples of each other. Here each component of b⃗2\vec b_2 is 3 times the matching one of b⃗1\vec b_1⁠: 6=3⋅26 = 3 \cdot 2⁠, 9=3⋅39 = 3 \cdot 3⁠, −18=3(−6)-18 = 3(-6)⁠.

Parallel lines are never skew and never meet in a single point, so (b) and (c) are ruled out. They are either coincident or distinct.

So the two lines are parallel. The point (1,1,−1)(1, 1, -1) of the first line is on the second line only if

1=2+6μ⇒μ=−161=−1+9μ⇒μ=29\begin{aligned} 1 = 2 + 6\mu &\Rightarrow \mu = -\frac{1}{6} \\ 1 = -1 + 9\mu &\Rightarrow \mu = \frac{2}{9} \end{aligned}
Why?

The first line passes through (1,1,−1)(1, 1, -1)⁠, the point i^+j^−k^\hat i + \hat j - \hat k at λ=0\lambda = 0⁠.

A general point of the second line is (2+6μ,−1+9μ,−1−18μ)(2 + 6\mu, -1 + 9\mu, -1 - 18\mu)⁠. Match its xx and yy coordinates with 1 and 1, and solve each for μ\mu⁠.

x:1=2+6μ6μ=1−2=−1μ=−16y:1=−1+9μ9μ=1+1=2μ=29\begin{aligned} x&: 1 = 2 + 6\mu \\ 6\mu &= 1 - 2 = -1 \\ \mu &= -\frac{1}{6} \\[4pt] y&: 1 = -1 + 9\mu \\ 9\mu &= 1 + 1 = 2 \\ \mu &= \frac{2}{9} \end{aligned}

These values differ, so the point is not on the second line, and the lines are not coincident.

Why?

One value of μ\mu would have to fit all three coordinates at once. Two different values mean that no point of the second line is (1,1,−1)(1, 1, -1)⁠. So the lines are parallel and distinct.

Answer: (d) parallel

Where this comes from

Chapter 11, Three Dimensional Geometry: Parallel and coincident lines (vector form).

Question type: Perpendicular or parallel

How to spot it: click to see

Type 7. Perpendicular or parallel: test the direction ratios

Spot it: "Show that the lines are perpendicular (or parallel)", or "find kk so that the lines are at right angles".

Here: 6i^+9j^−18k^=3(2i^+3j^−6k^)6\hat i + 9\hat j - 18\hat k = 3(2\hat i + 3\hat j - 6\hat k)⁠, so the directions are in proportion and the lines are parallel. The point (1,1,−1)(1, 1, -1) gives two different values of μ\mu on the second line, so they are not coincident.

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