Write the vector equation of the line whose Cartesian equations are $y = 2$ and $4x - 3z + 5 = 0$.

Write the vector equation of the line whose Cartes – Class 12 Mathematics

Miscellaneous Exercise 6A · Q42

y = 2

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The line is the intersection of the plane $y=2$ and the plane $4x-3z+5=0$ .

Finding a point: try $x=1$ : then $4(1)-3z+5=0\implies 3z=9\implies z=3$ . With $y=2$ , the point $(1,2,3)$ lies on the line (check: $y=2$ ✓ and $4(1)-3(3)+5=4-9+5=0$ ✓).

Finding the direction: the plane $y=2$ has normal $(0,1,0)$ and the plane $4x-3z+5=0$ has normal $(4,0,-3)$ . The line of intersection is perpendicular to both normals, so its direction is along their cross product: …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set ANNUAL2 marksMCQ
$4\hat i - \hat j + 2\hat k$
(a) $(4\hat i-\hat j-2\hat k)+\lambda(-2\hat i-\hat j+\hat k)$
(b) $(4\hat i-\hat j+2\hat k)+\lambda(2\hat i-\hat j+\hat k)$
(c) $(4\hat i-\hat j+2\hat k)+\lambda(-2\hat i-\hat j-\hat k)$
(d) $(4\hat i-\hat j+2\hat k)+\lambda(-2\hat i-\hat j+\hat k)$
›Reveal solutionSolution

Vector equation of a line through a point $A(\bar a)$ parallel to $\bar b$ is $\bar r=\bar a+\lambda\bar b$ .

The line passes through the point with position vector $\bar a = 4\hat i-\hat j+2\hat k$ and is parallel to $\bar b=-2\hat i-\hat j+\hat k$ .

The vector equation of a line through a fixed point $\bar a$ and parallel to $\bar b$ is

$\bar r=\bar a+\lambda \bar b$

Substituting,

$\bar r=(4\hat i-\hat j+2\hat k)+\lambda(-2\hat i-\hat j+\hat k)$

…

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CBSE 2024Set ANNUAL2 marks
$4\hat{i}-\hat{j}+2\hat{k}$
›Reveal solutionSolution

Vector equation: (point) + $\lambda$ (direction).

Line through position vector $\bar a=4\hat i-\hat j+2\hat k$ , parallel to $\bar b=-2\hat i-\hat j+\hat k$ .

…

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CBSE 2018Set ANNUAL2 marks
$4\hat{i} - \hat{j} + 2\hat{k}$
›Reveal solutionSolution

Vector equation of a line through a point $\bar a$ in the direction $\bar b$ is $\bar r = \bar a + \lambda \bar b$ .

The line passes through the point with position vector $\bar a = 4\hat i - \hat j + 2\hat k$ and has direction vector $\bar b = -2\hat i + \hat j + \hat k$ .

The vector equation of a line through a given point with position vector $\bar a$ along direction $\bar b$ is: …

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Q.Write the vector equation of the line whose Cartesian equations are $y = 2$ and $4x - 3z + 5 = 0$ .

Concept understanding — Vector Equation of a Line through a Point and Parallel to a Vector

Q.Write the vector equation of the line whose Cartesian equations are y=2y = 2y=2 and 4x−3z+5=04x - 3z + 5 = 04x−3z+5=0.

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Q.Write the vector equation of the line whose Cartesian equations are $y = 2$ and $4x - 3z + 5 = 0$ .

Unlock everything free for 14 days