Let a-k- b- and c-2-3k- be three given vectors- If r- is a vector such that r-b-c-b- and r-a-0- then the value of r-b- is

We have r⃗×b⃗=c⃗×b⃗ ⇒ (r⃗ c⃗)×b⃗=0⃗ ⇒r⃗ c⃗=λb⃗, λ≠ 0 ⇒r⃗=c⃗+λb⃗ Since r⃗.a⃗=0 ⇒ (c⃗+λb⃗).a⃗=0 ⇒λ =4 ⇒r⃗.b⃗=(c⃗+4b⃗).b⃗ =( 3î+6ĵ+3k̂).( î+ĵ) =3+6=9 ...

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Byju's Answer

Standard XII

Mathematics

Multiplication of a Vector by a Scalar

Let a⃗=-î-k̂,...

Question

Let $\to a = -^i -^k, \to b = -^i +^j and \to c =^i + 2^j + 3^k$ be three given vectors. If $\to r$ is a vector such that $\to r \times \to b = \to c \times \to b and \to r . \to a = 0,$ then the value of $\to r . \to b$ is

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Solution

We have
$\to r \times \to b = \to c \times \to b$
$\Rightarrow (\to r - \to c) \times \to b = \to 0$
$\Rightarrow \to r - \to c = λ \to b, λ \neq 0$
$\Rightarrow \to r = \to c + λ \to b$
Since $\to r . \to a = 0$
$\Rightarrow (\to c + λ \to b) . \to a = 0$
$\Rightarrow λ = 4$
$\Rightarrow \to r . \to b = (\to c + 4 \to b) . \to b$
$= (- 3^i + 6^j + 3^k) . (-^i +^j)$
$= 3 + 6 = 9$

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Let →a=−^i−^k, →b=−^i+^j and →c=^i+2^j+3^k be three given vectors. If →r is a vector such that →r×→b=→c×→b and →r.→a=0, then the value of →r.→b is

We have →r×→b=→c×→b ⇒(→r−→c)×→b=→0 ⇒→r−→c=λ→b, λ≠0 ⇒→r=→c+λ→b Since →r.→a=0 ⇒(→c+λ→b).→a=0 ⇒λ=4 ⇒→r.→b=(→c+4→b).→b =(−3^i+6^j+3^k).(−^i+^j) =3+6=9

Let $\to a = -^i -^k, \to b = -^i +^j and \to c =^i + 2^j + 3^k$ be three given vectors. If $\to r$ is a vector such that $\to r \times \to b = \to c \times \to b and \to r . \to a = 0,$ then the value of $\to r . \to b$ is