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Equal Vectors

The vectors ...

Question

The vectors $a = x i - 2 j + 5 k$ and $b = i + y j - z k$ are collinear if

x = 1, y = - 2, z = - 5

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x = 1 / 2, y = - 4, z = - 10

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x = - 1 / 2, y = 4, z = 10

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x = - 1, y = 2, z = 5.

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Solution

The correct options are
A $x = 1, y = - 2, z = - 5$
B $x = 1 / 2, y = - 4, z = - 10$
C $x = - 1 / 2, y = 4, z = 10$
D $x = - 1, y = 2, z = 5.$
The given vectors are collinear if $a = k b$ for some $k$
i.e $x i - 2 j + 5 k = k i + k y j - z k k$
$\Rightarrow (x - k) i + (- 2 - k y) j + (5 + z k) k = 0$
Hence $x = k, y = - 2 / k$ and
$z = - 5 / k .$ Taking $k = 1, 1 / 2, - 1 / 2$ and $- 1$ we get the values given by $(a), (b), (c)$ and $(d),$ respectively

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