If a=2i+k, b=i+j+k, c=4i-3j+7k then the vector r satifying r × b=c×b and r.a=0 is

r×b=c×b ⇒ a×(r×b)=a×(c×b)(a·b) r-0=(a·b) c-(a·c) br=c-a·ca·b br=4i-3j+7k-5 (i+j+k) =-i-8j+2k

If a=2i+k, b=i+j+k, c=4i-3j+7k then the vector r satifying r × b=c×b and r.a=0 is

r×b=c×b ⇒ a×(r×b)=a×(c×b)(a·b) r-0=(a·b) c-(a·c) br=c-a·ca·b br=4i-3j+7k-5 (i+j+k) =-i-8j+2k

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If $a = 2 i + k, b = i + j + k, c = 4 i - 3 j + 7 k$ then the vector $r$ satifying $r \times b = c \times b$ and $r . a = 0$ is

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$i + 8 j + 2 k$

$i - 8 j + 2 k$

$i - 8 j - 2 k$

$- i - 8 j + 2 k$

answer is D.

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Detailed Solution

$\begin{array}{rcl} r \times b & = & c \times b \Rightarrow a \times (r \times b) = a \times (c \times b) \\ (a \cdot b) r - 0 & = & (a \cdot b) c - (a \cdot c) b \\ r & = & c - \frac{a \cdot c}{a \cdot b} b \\ r & = & 4 i - 3 j + 7 k - 5 (i + j + k) \\ = & - i - 8 j + 2 k \end{array}$