Question

$\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{is}\mathrm{the}\mathrm{vector}\mathrm{equation}\mathrm{of}a\mathrm{line}\mathrm{which}\mathrm{passes}\mathrm{through}\mathrm{point}P(2,-1,4)\mathrm{and}\mathrm{having}\mathrm{the}\mathrm{direction}\mathrm{along}\mathrm{the}\mathrm{vector}\hat{i}+3\hat{j}-2\hat{k}.$

• A.
  $(2+\lambda )\hat{i}+(-1+3\lambda )\hat{j}+(4-2\lambda )\hat{k}$
• B. $(2-\lambda )\hat{i}+(-1-3\lambda )\hat{j}+(4+2\lambda )\hat{k}$
• C. $(1+2\lambda )\hat{i}+(3-\lambda )\hat{j}+(-2+4\lambda )\hat{k}$
• D. $\lambda \hat{i}+3\lambda \hat{j}-2\lambda \hat{k}$

Answer 1

The correct option is A

$(2+\lambda )\hat{i}+(-1+3\lambda )\hat{j}+(4-2\lambda )\hat{k}$
The vector equation of a line passing through a point 'a' and in the direction of vector 'b' is given by
$\overrightarrow{V}=\overrightarrow{a}+\lambda \overrightarrow{b}$Here,
$\overrightarrow{a}=2\hat{i}-\hat{j}+4\hat{k}$
and $\overrightarrow{b}=\hat{i}+3\hat{j}-2\hat{k}$
$\overrightarrow{V}=2\hat{i}-\hat{j}+4\hat{k}+\lambda \left(\hat{i}+3\hat{j}-2\hat{k}\right)=\left(2+\lambda \right)\hat{i}+\left(-1+3\lambda \right)\hat{j}+\left(4-2\lambda \right)\hat{k}$