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Question

What vector when added to $$(2\hat i-2\hat j+\hat k)$$ and $$(2\hat i-\hat k)$$ will give a unit vector along negative $$y-$$axis?


Solution

$$(2\hat i - 2\hat j + \hat k) + (2\hat i - \hat k) = (4\hat i - 2\hat j + 0\hat k)$$
Now we have to find a vector which when added to the above vector gives along negative y-axis i.e. $$-\hat j$$.

Let the required vector be $$x\hat i + y\hat j + z\hat k$$
$$\begin{array}{l} \Rightarrow (4\hat { i } -2\hat { j } +0\hat { k } )+(x\hat { i } +y\hat { j } +z\hat { k } )=-\hat { j }  \\ \Rightarrow (4+x)\hat { i } +(-2+y)\hat { j } +(0+z)\hat { k }  \\ 4+x=0\Rightarrow x=-4 \\ -2+y=-1\Rightarrow y=1 \\ 0+z=0\Rightarrow z=0 \end{array}$$

Therefore the required vector is $$-4\hat i + \hat j$$

Physics

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