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Question Number 54419 by gunawan last updated on 03/Feb/19

$$\mathrm{If}\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k}\mathrm{and}\mathbf{c}\mathrm{is}\mathrm{a}\mathrm{unit}$$$$\mathrm{vector}\mathrm{\perp }\mathrm{to}\mathrm{the}\mathrm{vector}\mathbf{a}\mathrm{and}\mathrm{coplanar}$$$$\mathrm{with}\mathbf{a}\mathrm{and}\mathbf{b},\mathrm{then}\mathrm{a}\mathrm{unit}\mathrm{vector}\mathbf{d}\mathrm{\perp }\mathrm{to}$$$$\mathrm{both}\mathbf{a}\mathrm{and}\mathbf{c}\mathrm{is}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Feb/19

$$c=x\mathit{a}+y\mathit{b}$$$$\mathit{c}=x(i+j-k)+y(i-j+k)$$$$\mathit{c}=i(x+y)+j(x-y)+k(-x+y)$$$$\mathit{c}.\mathit{a}=0$$$$[i(x+y)+j(x-y)+k(-x+y)].[i+j-k]=0$$$$(x+y)+(x-y)+(-x+y)(-1)=0$$$$x+y+x-y+x-y=0$$$$3x-y=0.....\left(1\right)eqn$$$$\mathit{c}=i(x+y)+j(x-y)+k(-x+y)$$$$\mathit{c}=i(x+3x)+j(x-3x)+k(-x+3x)$$$$\mathit{c}=i\left(4x\right)+j(-2x)+k\left(2x\right)$$$$given\sqrt{16x^2+4x^2+4x^2}=1$$$$\sqrt{24x^2}=1x=\pm \frac{1}{\sqrt{24}}$$$$considering+vevalueonly...x=\frac{1}{\sqrt{24}}y=\frac{3}{\sqrt{24}}$$$$\mathit{c}=\mathit{i}(\mathit{x}+\mathit{y})+\mathit{j}(\mathit{x}-\mathit{y})+\mathit{k}(-\mathit{x}+\mathit{y})$$$$\mathit{c}=\mathit{i}\left(\frac{4}{\sqrt{24}}\right)+\mathit{j}\left(\frac{-2}{\sqrt{24}}\right)+\mathit{k}\left(\frac{2}{\sqrt{24}}\right)$$$$\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{r}\mathit{i}\mathit{n}\mathit{g}-\mathit{v}\mathit{e}\mathit{s}\mathit{i}\mathit{g}\mathit{n}...$$$$\mathit{c}=\mathit{i}\left(\frac{-4}{\sqrt{24}}\right)+\mathit{j}\left(\frac{2}{\sqrt{24}}\right)+\mathit{k}\left(\frac{-2}{\sqrt{24}}\right)$$$$$$$$\mathit{d}=\frac{\mathit{a}\times \mathit{c}}{\mid \mathit{a}\times \mathit{c}\mid }now\mathit{a}\times \mathit{c}$$$$\left[\mathit{i}\mathit{j}\mathit{k}\right]$$$$[11-1]$$$$\left[\frac{4}{\sqrt{24}}\frac{-2}{\sqrt{24}}\frac{2}{\sqrt{24}}\right]$$$$nowyousolvetheremaining...$$

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