Let A-2-k-B-k- and C-4-3-7k-Determine a vector R satisfying R-A-0 and R-B-C-B

R×B=C×B⇒(R C)×B=0 ⇒ nbsp; R=C+λB R.A⇒C.A+λB.A=0 ⇒(4î 3ĵ+7k̂)(2î+k̂)+λ(î+ĵ+k̂)(2î+k̂)=0 ⇒(8+7)+λ(2+1)=0 ⇒ 15+3λ=0 ⇒λ= 5 ...

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Byju's Answer

Standard XII

Mathematics

Dot Product of Two Vectors

Let A=2î+k̂...

Question

Let $\to A = 2^i +^k, \to B =^i +^j +^k$ and $\to C = 4^i - 3^j + 7^k$ .Determine a vector $\to R$ satisfying $\to R . \to A = 0$ and $\to R \times \to B = \to C \times \to B$

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Solution

$\to R \times \to B = \to C \times \to B \Rightarrow (\to R - \to C) \times \to B = 0$
$\Rightarrow \to R = \to C + λ \to B$
$\to R . \to A \Rightarrow \to C . \to A + λ \to B . \to A = 0$
$\Rightarrow (4^i - 3^j + 7^k) (2^i +^k) + λ (^i +^j +^k) (2^i +^k) = 0$
$\Rightarrow (8 + 7) + λ (2 + 1) = 0$
$\Rightarrow 15 + 3 λ = 0$
$\Rightarrow λ = - 5$
$∴ \to R = \to C - 5 \to B$
$= (4^i - 3^j + 7^k) - 5 (^i +^j +^k)$
$= 4^i - 3^j + 7^k - 5^i - 5^j - 5^k$
$= -^i - 8^j + 2^k$

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Let →A=2^i+^k,→B=^i+^j+^k and →C=4^i−3^j+7^k.Determine a vector →R satisfying →R.→A=0 and →R×→B=→C×→B

→R×→B=→C×→B⇒(→R−→C)×→B=0⇒→R=→C+λ→B→R.→A⇒→C.→A+λ→B.→A=0⇒(4^i−3^j+7^k)(2^i+^k)+λ(^i+^j+^k)(2^i+^k)=0⇒(8+7)+λ(2+1)=0⇒15+3λ=0⇒λ=−5∴→R=→C−5→B=(4^i−3^j+7^k)−5(^i+^j+^k)=4^i−3^j+7^k−5^i−5^j−5^k=−^i−8^j+2^k

Let $\to A = 2^i +^k, \to B =^i +^j +^k$ and $\to C = 4^i - 3^j + 7^k$ .Determine a vector $\to R$ satisfying $\to R . \to A = 0$ and $\to R \times \to B = \to C \times \to B$