p-south-1.amazonaws.com/pyq-question-images/batch-001/5b438983d94f.webp" data-orsrc="https://uploads-pracbee-ramanuj.s3.ap-south-1.amazonaws.com/pyq-question-images/batch-001/5b438983d94f.webp" loading="lazy" style="max-width:100%;height:auto;display:block;margin:0 auto;max-height:40vh;vertical-align:baseline" alt="TG EAPCET 2024 (Online) 9th May Morning Shift Mathematics - Vector Algebra Question 121 English Explanation">

$$ A(\hat{i}-2 \hat{j}+\hat{k}) $$

$$ B(2 \hat{i}-\hat{j}+\hat{k}) \quad C(\hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) $$

Position vector of

$$ \begin{aligned} D & =\frac{2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}+\hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}}{2} \\\\ & =\frac{3}{2}(\hat{\mathbf{i}}-\hat{\mathbf{k}}) \end{aligned} $$

Position vector of

$$ \begin{aligned} E & =\frac{\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}+\hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}}{2} \\\\ & =\frac{1}{2}(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \\\\ \therefore \mathrm{DE}= & \frac{1}{2}(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}})-\frac{3}{2}(\hat{\mathbf{i}}-\hat{\mathbf{k}}) \\\\ = & -\frac{1}{2} \hat{\mathbf{i}}-\frac{3}{2} \hat{\mathbf{j}}+\hat{\mathbf{k}} \end{aligned} $$

$\Rightarrow$ Unit vector along

$\begin{aligned} \mathrm{DE} & =\frac{\left(-\frac{1}{2} \hat{\mathbf{i}}-\frac{3}{2} \hat{\mathbf{j}}+\hat{\mathbf{k}}\right)}{\sqrt{\left(-\frac{1}{2}\right)^2+\left(\frac{-3}{2}\right)^2+(1)^2}} \\ & =\frac{1}{\sqrt{14}}(-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})\end{aligned}$

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