The vector [(i-j)×(j-k)]×(i +5k) is

a=i-j b=j-k c=i+5k ⇒(a×b)×c=(c·a) b-(c·b) a =(1) b-(-5) a =5a+b=5i-5j+j-k =5i-4j-k

The vector [(i-j)×(j-k)]×(i +5k) is

a=i-j b=j-k c=i+5k ⇒(a×b)×c=(c·a) b-(c·b) a =(1) b-(-5) a =5a+b=5i-5j+j-k =5i-4j-k

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The vector $[(i - j) \times (j - k)] \times (i + 5 k)$ is

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$5 i + 4 j - k$

$4 i - 5 j - k$

$5 i - 4 j - k$

$3 i - 2 j + 5 k$

answer is A.

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Detailed Solution

$\begin{array}{rcl} a & = & i - j b = j - k c = i + 5 k \\ \Rightarrow & (a \times b) \times c = (c \cdot a) b - (c \cdot b) a \\ = & (1) b - (- 5) a \\ = & 5 a + b = 5 i - 5 j + j - k \\ = & 5 i - 4 j - k \end{array}$

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The vector [(i-j)×(j-k)]×(i +5k) is

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The vector $[(i - j) \times (j - k)] \times (i + 5 k)$ is