Let $L$ be the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$. If $L$ makes an angles $\alpha$ with the positive $x$-axis, then $\cos \alpha$ equals

JEE Main · 2007 · 29 Apr 2007 · Offline · Three Dimensional Geometry

Let

L

be the line of intersection of the planes

2 x + 3 y + z = 1

and

x + 3 y + 2 z = 2

. If

L

makes an angles

α

with the positive

x

-axis, then

cos α

equals

(A) $\frac{1}{3}$ ✓ Correct
(B) $\frac{1}{2}$
(C)1
(D) $\frac{1}{2}$

Answer: (A)

Solution

The line of intersection runs along the direction perpendicular to both plane normals,

n_{1} = (2, 3, 1)

and

n_{2} = (1, 3, 2)

. Their cross product is

n_{1} \times n_{2} = (6 - 3, 1 - 4, 6 - 3) = (3, - 3, 3)

, which is parallel to

(1, - 1, 1)

. Normalising, the direction cosines are

\pm \frac{1}{3}

apiece, so the cosine of the angle the line makes with the positive

x

-axis is

\frac{1}{3}

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option

Practise more: all JEE Main PYQs on Three Dimensional Geometry →

More JEE Main questions on Three Dimensional Geometry

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .
Then the length of the projection of the vector …
JEE Main 2026View solution →
Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
JEE Main 2026View solution →
Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point $(2, 6, 7)$ and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point $(4, 3, 5)$ . If the line $L_{3}$ is parallel to the vector …
JEE Main 2026View solution →
Let $P (α, β, γ)$ be the point on the line
$\frac{x - 1}{2} = \frac{y + 1}{- 3} = z$ at a distance $414$ from the point $(1, - 1, 0)$ and nearer to the origin. Then the shortest distance, between the lines …
JEE Main 2026View solution →
If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to
JEE Main 2026View solution →
Let L be the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z + 3}{6}$ and let S be the set of all points $(a, b, c)$ on L, whose distance from the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z - 9}{0}$ along the line L is 7. Then $\sum_{(a, b, c) \in S} (a + b + c)$ is equal to :
JEE Main 2026View solution →