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Distance between Two Parallel Planes

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Question

What is the value of $λ$ for which the lines $\frac{x - 1}{2} = \frac{y - 3}{5} = \frac{z - 1}{λ}$ and $\frac{x - 2}{3} = \frac{y + 1}{- 2} = \frac{z}{2}$ are perpendicular to each other?

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Solution

The given lines are perpendicular to each other. If the sum of the products of their direction ratios is zero.

That is:

$2 \times 3 + 5 \times - 2 + λ \times 2 = 0$

$λ = 2$

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Q. The number of distinct real value of

λ

for which the lines

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\frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2}

are coplanar is:

Q. Show that the lines

\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1}

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Q. Show that the lines

\frac{x - 3}{2} = \frac{y + 1}{- 3} = \frac{z - 2}{4}

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Q. The number of distinct real values of

λ

for which the lines

\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{λ^{2}}

\frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2}

are coplanar is:

Q. The value of

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