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Area Method to Find Condition for Co-Linearity

Let a⃗=î+ĵ ...

Question

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

A

-^i +^j +^k

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B

- 3^i -^j +^k

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C

3^i +^j -^k

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D

^i -^j -^k

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Solution

The correct option is B $3^i +^j -^k$
$\to a =^i +^j$ & $^b =^2 i -^k \to r \times \to a = \to b \times \to a \Rightarrow (\to r - \to b) \times \to a = 0 \Rightarrow \to r = \to b + λ \to a$
Similarly $\to r \times \to b = \to a \times \to b \Rightarrow \to r = \to a + μ \to b$
Substitute the vector $\to a$ & $\to b$ in (i) & (ii) and equating we get
$2^i -^k + λ (^i +^j) =^i +^j + μ (2^i -^k) \Rightarrow 2 + λ = 1 + 2 μ, λ = 1, μ = 1 ∴$ Point of intersection is $3^i +^j -^k$

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Similar questions

\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

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

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

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

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

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

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

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

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

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

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

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Area Method to Find Condition for Co-Linearity

Standard XII Mathematics

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