Find the value of - which will make the vectors a- b- and c- coplanar- where a-2 -k- b-2 -3 k- and c-3 -5 k-

A̅,b̅,c̅ are coplanar⇒ 2 amp; 1 amp;1 1 amp;2 amp; 3 3 amp; λ amp;5 =0 ⇒ 2(10 3 λ) + 1(5+9)+1( λ 6)=0 ⇒ 7 λ = 28 ⇒λ = 4 ...

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Condition for Two Lines to be on the Same Plane

Find the valu...

Question

Find the value of $λ$ which will make the vectors $\to a, \to b and \to c$ coplanar, where $\to a = 2^i -^j +^k, \to b =^i + 2^j - 3^k and \to c = 3^i - λ^j + 5^k$ .

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Solution

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\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) =

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

\to c = 3^i +^j

\to a + λ \to b

\to c

λ

\to a \cdot (\to b \times \to c)

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

\to c = 3^i +^j + 2^k .

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

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

\to a \cdot (\to b \times \to c)

\to A = 3^i - 2^j +^k, \to B =^i - 3^j + 5^k

\to C = 2^i -^j + 4^k

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