Question

Q5 slove and also solve how to get x , y,z values of vector

​Q5. Scalar products of a vector with the vectors $3\hat{\mathrm{i}}-5\hat{\mathrm{k}},2\hat{\mathrm{i}}+7\mathrm{j}\mathrm{and}\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ are respectively – 1, 6 and 5 find the vector.

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{required}\mathrm{vector}=\mathrm{xi}+\mathrm{yj}+\mathrm{zk} \\ \left(\mathrm{xi}+\mathrm{yj}+\mathrm{zk}\right)\left(3\mathrm{i}-5\mathrm{k}\right)=-1 \\ 3\mathrm{x}-5\mathrm{z}=-1 \\ \Rightarrow 3\mathrm{x}=5\mathrm{z}-1 \\ \Rightarrow \mathrm{x}=\frac{5\mathrm{z}-1}{3}...\left(1\right) \\ \left(\mathrm{xi}+\mathrm{yj}+\mathrm{zk}\right)\left(2\mathrm{i}+7\mathrm{j}\right)=6 \\ 2\mathrm{x}+7\mathrm{y}=6 \\ 2\mathrm{x}=6-7\mathrm{y} \\ 2\left(\frac{5\mathrm{z}-1}{3}\right)=6-7\mathrm{y} \\ 2\left(5\mathrm{z}-1\right)=18-21\mathrm{y} \\ 5\mathrm{z}-1=\frac{18-21\mathrm{y}}{2} \\ 5\mathrm{z}=\frac{18-21\mathrm{y}}{2}+1 \\ 5\mathrm{z}=\frac{20-21\mathrm{y}}{2} \\ \mathrm{z}=\frac{20-21\mathrm{y}}{10}...\left(2\right) \\ \mathrm{And}\left(\mathrm{xi}+\mathrm{yj}+\mathrm{zk}\right)\left(\mathrm{i}+\mathrm{j}+\mathrm{k}\right)=5 \\ \mathrm{x}+\mathrm{y}+\mathrm{z}=5 \\ \frac{5\left(\frac{20-21\mathrm{y}}{10}\right)-1}{3}+\mathrm{y}+\frac{20-21\mathrm{y}}{10}=5\left[\mathrm{using}\left(1\right)\mathrm{and}\left(2\right)\right] \\ \frac{{\displaystyle \frac{20-21\mathrm{y}}{2}}-1}{3}+\mathrm{y}+\frac{20-21\mathrm{y}}{10}=5 \\ \frac{{\displaystyle \frac{20-21\mathrm{y}-2}{2}}}{3}+\mathrm{y}+\frac{20-21\mathrm{y}}{10}=5 \\ \frac{18-21\mathrm{y}}{6}+\mathrm{y}+\frac{20-21\mathrm{y}}{10}=5 \\ \frac{90-105\mathrm{y}+30\mathrm{y}+60-63\mathrm{y}}{30}=5 \\ 150-138\mathrm{y}=150 \\ -138\mathrm{y}=0 \\ \boxed{\mathbf{y}\mathbf{=}\mathbf{0}} \\ \mathrm{and}\mathrm{z}=\frac{20-21\left(0\right)}{10}=2 \\ \boxed{\mathrm{z}=2} \\ \mathrm{and}\mathrm{x}=\frac{5\left(2\right)-1}{3}=\frac{9}{3} \\ \boxed{\mathbf{x}\mathbf{=}\mathbf{3}} \\ \mathrm{So}\mathrm{required}\mathrm{vector}\mathrm{is}: \\ =3\mathrm{i}+0\mathrm{j}+2\mathrm{k} \\ =3\mathrm{i}+2\mathrm{k} \end{gathered}$$
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