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Show that the line whose vector equation is is parallel to the plane whose vector equation is …
Show that the line whose vector equation is $\vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k})$ is parallel to the plane whose vector equation is $\vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5$. Also find the distance between them. [Application]
Given: \vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k}), \quad \vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5
To show: \vec{r} \text{ is parallel to the plane}
Answer
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From ISC 2025 Specimen Mathematics Paper 1, question 17(i).