Let a- - - - - and b- - 2- - k- The point of intersection of the lines r-a- - b-a- and r-b- - a-b- is

The correct option is C 3î + ĵ k̂ Let point of intersection =r⃗ =mî +nĵ +pk̂ ⇒ r⃗×a⃗ =| î #160; amp; ĵ #160; amp; k̂ #160; ...

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Byju's Answer

Standard XII

Mathematics

Skew Lines

Let a⃗ = î ...

Question

Let $\to a =^i +^j$ and $\to b = 2^i -^k$ . The point of intersection of the lines $\to r \times \to a = \to b \times \to a$ and $\to r \times \to b = \to a \times \to b$ is

-^i +^j + 2^k

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3^i -^j +^k

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3^i +^j -^k

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^i -^j -^k

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Solution

The correct option is C $3^i +^j -^k$
Let point of intersection $= \to r = m^i + n^j + p^k$
$\Rightarrow \to r \times \to a = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k m & n & p 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow^i (0 - p) -^j (0 - p) +^k (m - n)$
$\Rightarrow \to b \times \to a = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 0 & - 1 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow^i (1) -^j (1) +^k (2)$
$\Rightarrow^i -^j + 2^k$
$\Rightarrow \to r \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k m & n & p 2 & 0 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow^i (- n) -^j (- m - 2 p) +^k (- 2 n)$
$\Rightarrow - n^i +^j (m + 2 p) +^k (- 2 n)$
$\to a \times \to b = - (\to b \times \to a) = -^i +^j - 2^k$
$∵ (\to r \times \to a) = (\to b \times \to a)$
$\Rightarrow^i (- p) +^j (p) +^k (m - n) =^i +^j (- 1) +^k (2)$
$\Rightarrow p = - 1 \to (1) m - n = 2 \to (2)$
and $(\to r \times \to a) = (\to b \times \to a)$
$\Rightarrow^i (- n) +^j (m + 2 p) +^k (- 2 n) =^i (- 1) +^j (1) +^k (- 2)$
$\Rightarrow n = 1 \to (3) m + 2 p = 1 \to (4)$
$\Rightarrow m - 2 = 1 \Rightarrow m = 3$
$n = 1 p = - 1$
$∴ 3^i +^j -^k$ is the ans

Suggest Corrections

Similar questions

Q. If

\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

and

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

If $\to r \times \to b = \to c \times \to b$ and $\to r . \to a = 0$ where $\to a = 2^i + 3^j -^k, \to b = 3^i -^j +^k$ and $\to c =^i +^j + 3^k$ , then $\to r$ is equal to

Q. Let

\to a =^i +^j

\to b = 2^i +^j

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

Q. Let

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

. Then, vector

\to r

satisfying the equations

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

and

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

Q. If

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

\to b = 3^i + 2^j +^k

\to c =^i -^j - 4^k

\to d =^i + 2^j -^k

, then

(\to a \times \to b) \times (\to c \times \to d) =

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Let →a=^i+^j and →b=2^i−^k. The point of intersection of the lines →r×→a=→b×→a and →r×→b=→a×→b is

Let $\to a =^i +^j$ and $\to b = 2^i -^k$ . The point of intersection of the lines $\to r \times \to a = \to b \times \to a$ and $\to r \times \to b = \to a \times \to b$ is