Find the vector equation of the line which is passing through the point (2,-3,5) and parallel to the vector \(3\hat{i}+4\hat{j}-2\hat{k}\).

Category: QuestionsFind the vector equation of the line which is passing through the point (2,-3,5) and parallel to the vector \(3\hat{i}+4\hat{j}-2\hat{k}\).
Editor">Editor Staff asked 11 months ago

Find the vector equation of the line which is passing through the point (2,-3,5) and parallel to the vector \(3\hat{i}+4\hat{j}-2\hat{k}\).
 
(a) \((2+3λ) \hat{i}+(4λ+3) \hat{j}+(5-λ)\hat{k}\)
 
(b) \((9+3λ) \hat{i}+(λ-3) \hat{j}+(5-2λ)\hat{k}\)
 
(c) \((2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}\)
 
(d) \((7+λ) \hat{i}+(4λ+3) \hat{j}+(5-2λ)\hat{k}\)
 
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This key question is from Three Dimensional Geometry topic in division Three Dimensional Geometry of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct answer is (c) (2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}
 
To explain: Given that the line is passing through the point (2,-3,5). Therefore, the position vector of the line is \vec{a}=2\hat{i}-3\hat{j}+5\hat{k}.
 
Also given that, the line is parallel to a vector \vec{b}=3\hat{i}+4\hat{j}-2\hat{k}.
 
We know that, the equation of line passing through a point and parallel to vector is given by \vec{r}=\vec{a}+λ\vec{b}, where λ is a constant.
 
∴\vec{r}=2\hat{i}-3\hat{j}+5\hat{k}+λ(3\hat{i}+4\hat{j}-2\hat{k})
 
=(2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}