Let a-2-k-b- and c-k- be three given vectors- If r is a vector such that r-a-c-a and r-b-0- then r-a is equal to

R×a=c×a r×a c×a=0 (r c)×a=0 ∴r c=λa r=λa+c r·b=λa·b+c·b=0 ⇒λ (1 2)+2=0 ⇒λ =2 ∴r=2a+c r·a=2|a|^2+a·c =2(1+4+1)+(1 2+1) =12 nbsp; ...

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Byju's Answer

Standard XII

Mathematics

Linear Combination of Vectors

Let a=î+2ĵ-k̂...

Question

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

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Solution

$\to r \times \to a = \to c \times \to a$
$\to r \times \to a - \to c \times \to a = 0$
$(\to r - \to c) \times \to a = 0$
$∴ \to r - \to c = λ \to a$
$\to r = λ \to a + \to c$

$\to r \cdot \to b = λ \to a \cdot \to b + \to c \cdot \to b = 0$
$\Rightarrow λ (1 - 2) + 2 = 0$
$\Rightarrow λ = 2$
$∴ \to r = 2 \to a + \to c$
$\to r \cdot \to a = 2 | \to a |^{2} + \to a \cdot \to c$
$= 2 (1 + 4 + 1) + (1 - 2 + 1)$
$= 12$

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

→r×→a=→c×→a →r×→a−→c×→a=0 (→r−→c)×→a=0 ∴→r−→c=λ→a →r=λ→a+→c →r⋅→b=λ→a⋅→b+→c⋅→b=0 ⇒λ(1−2)+2=0 ⇒λ=2 ∴→r=2→a+→c →r⋅→a=2|→a|2+→a⋅→c =2(1+4+1)+(1−2+1) =12

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