Let a¯=2i¯+3j¯-k¯, b¯=3i¯-j¯+k¯ andc¯=i¯+j¯+3k¯.If r¯×b¯=c¯×b¯, r¯.a¯=0, then r¯=

r×b=c×b r·a=0 a×(r×b)=a×(c×b) (a·b) r-0=(a·b) c-(a·c) b r=c-a·ca·b b r=(i+j+3k)-2+3-36-3-1 (3i-j+k) =-2i+2j+2k

Let a¯=2i¯+3j¯-k¯, b¯=3i¯-j¯+k¯ andc¯=i¯+j¯+3k¯.If r¯×b¯=c¯×b¯, r¯.a¯=0, then r¯=

r×b=c×b r·a=0 a×(r×b)=a×(c×b) (a·b) r-0=(a·b) c-(a·c) b r=c-a·ca·b b r=(i+j+3k)-2+3-36-3-1 (3i-j+k) =-2i+2j+2k

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Let $\bar{a} = 2 \bar{i} + 3 \bar{j} - \bar{k}, \bar{b} = 3 \bar{i} - \bar{j} + \bar{k}$ and
$\bar{c} = \bar{i} + \bar{j} + 3 \bar{k} .$ If $\bar{r} \times \bar{b} = \bar{c} \times \bar{b}, \bar{r} . \bar{a} = 0, then \bar{r} =$

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$- \bar{i} + \bar{j} + \bar{k}$

$2 (- \bar{i} + \bar{j} + \bar{k})$

$- 5 \bar{i} + \bar{j} + \bar{k}$

$- \bar{i} + 5 \bar{j} + \bar{k}$

answer is B.

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Detailed Solution

$r \times b = c \times b r \cdot a = 0 a \times (r \times b) = a \times (c \times b) (a \cdot b) r - 0 = (a \cdot b) c - (a \cdot c) b r = c - (\frac{a \cdot c}{a \cdot b}) b r = (i + j + 3 k) - (\frac{2 + 3 - 3}{6 - 3 - 1}) (3 i - j + k) = - 2 i + 2 j + 2 k$