The cosine of the angle included between the lines $\overline{\mathrm{r}}=(2 \hat{\imath}+\hat{\jmath}-2 \hat{k})+\lambda(\hat{\imath}-2 \hat{\jmath}-2 \hat{k})$ and $\overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath}+3 \hat{k})+\mu(3 \hat{\imath}+2 \hat{\jmath}-6 \hat{k})$ where $\lambda, \mu \in…

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The cosine of the angle included between the lines $\overline{\mathrm{r}}=(2 \hat{\imath}+\hat{\jmath}-2 \hat{k})+\lambda(\hat{\imath}-2 \hat{\jmath}-2 \hat{k})$ and $\overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath}+3 \hat{k})+\mu(3 \hat{\imath}+2 \hat{\jmath}-6 \hat{k})$ where $\lambda, \mu \in \mathrm{R}$ is

Aicharya · Accepted Answer

Given lines, $\begin{aligned} & r=(2 \hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-2 \hat{j}-2 \hat{k}) \ & r=(\hat{i}+\hat{j}+3 \hat{k})+\mu(3 \hat{i}+2 \hat{j}-6 \hat{k}) \end{aligned}$ Here, $\mathrm{b}_1=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\begin{aligned} & \mathrm{b}_2=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-6 \hat{\mathrm{k}} \ & \therefore \cos \theta=\frac{\mathrm{b}_1 \cdot \mathrm{b}_2}{\left|\mathrm{~b}_1\right|\left|\mathrm{b}_2\right|} \ & =\frac{(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}) \cdot(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-6 \hat{\mathrm{k}})}{(\sqrt{1+4+4})(\sqrt{9+4+36})} \ & \cos \theta=\frac{3-4+12}{3 \times 7} \ & =\frac{11}{21} \end{aligned}$

The cosine of the angle included between the lines $\overline{\mathrm{r}}=(2 \hat{\imath}+\hat{\jmath}-2…

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