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Vector Addition

Let r⃗= a⃗+...

Question

Let $\to r = \to a + λ \to l$ and $\to r = \to b + μ \to m$ be two lines in space where $\to a = 5^i +^j + 2^k,$ $\to b = -^i + 7^j + 8^k,$ $\to l = - 4^i + 7^j -^k$ and $\to m = - 2^i - 5^j - 7^k$ then the position vector of a point which lies on both of these lines, is

A

^i + 2^j +^k

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B

2^i +^j +^k

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C

^i + 2^j + 2^k

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D

Non existent as the lines are skew

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Solution

The correct option is D Non existent as the lines are skew
We need to compare the components of the point in both the lines.
Thus, $5 - 4 λ = - 1 - 2 μ$ ... $(1)$
Similarly, $1 + 7 λ = 7 - 5 μ$ ... $(2)$
and $2 - λ = 8 - 7 μ$ ... $(3)$

We solve any two equations simultaneously and substitute the value in the third equation.
Multipying first equation by $5$ and second by $2$ and subtracting both, we get
$30 - 20 λ = - 12 + 14 λ, \Rightarrow 34 λ = 42, λ = \frac{21}{17}$
$μ = \frac{6 - \frac{84}{17}}{- 2} = \frac{- 9}{17}$

$L . H . S$ . of third equation = $2 - \frac{21}{17} = \frac{13}{17}$
$R . H . S$ . of third equation = $8 + 7 \times \frac{9}{17} \neq \frac{13}{17}$

Hence the lines are skew.

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Similar questions

\to r = \to a + λ \to l

\to r = \to b = μ \to m

be two lines in space where

\to a = 5^i +^j + 2^k, \to b =^i + 7^j + 8^k, \to l = - 4^i +^j -^k a n d \to m = 2^i - 5^j - 7^k

then the p.v. of a point which lies on both of these lines, is

\to r = \to a + λ \to l

\to r = \to b + μ \to m

be two lines in space where

\to a = 5^i +^j + 2^k

\to b = -^i + 7^j + 8^k

\to l = 4^i +^j -^k] a n d \to m = 2^i - 5^j - 7^(k

then the p.v. of a point which lies on both of these lines, is

\to a = 3^i - 2^j +^k,

\to b = 2^i - 4^j - 3^k, \to c = -^i + 2^j + 2^k

\to a + \to b + \to c = 4^i - 4^j

\to a =^i -^j + 2^k, \to b = 2^i + 3^j +^k, \to c =^i -^k

, then magnitude of

\to a + 2 \to b - 3 \to c

\sqrt{78}

\to a =^i - 2^j +^k, \to b =^j +^k

, then the point of intersection of lines

\to r \times \to a = \to b \times \to a

\to r \times \to b = \to a \times \to b

\to a =^i +^j

\to b = 2^i -^k

. The point of intersection of the lines

\to r \times \to a = \to b \times \to a

\to r \times \to b = \to a \times \to b

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