Direction Cosines and Direction Ratios

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JEE Main 2014

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^{2} = m^{2} + n^{2}$ is

\frac{π}{6}

(B)

\frac{π}{2}

(C)

\frac{π}{3}

(D)

\frac{π}{4}

JEE Main 2010

LEVELBoard

A line $A B$ in three-dimensional space makes angles $4 5^{\circ}$ and $12 0^{\circ}$ with the positive $x$ -axis and the positive $y$ -axis respectively. If $A B$ makes an acute angle $θ$ with the positive $z$ -axis, then $θ$ equals

(A)

4 5^{\circ}

(B)

6 0^{\circ}

(C)

7 5^{\circ}

(D)

3 0^{\circ}

JEE Main 2007

LEVELJEE Main

If a line makes an angle of $π /4$ with the positive directions of each of $x$ -axis and $y$ -axis, then the angle that the line makes with the positive direction of the $z$ -axis is

(A)

π /4

(B)

π /2

(C)

π /6

(D)

π /3

JEE Main 2004

LEVELJEE Main

A line makes the same angle $θ$ , with each of the $x$ and $z$ axis. If the angle $β$ , which it makes with $y$ -axis, is such that $sin^{2} β = 3 sin^{2} θ$ , then $cos^{2} θ$ equals

(A)

2/5

(B)

1/5

(C)

3/5

(D)

2/3

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Main

The angle between the straight lines, whose direction cosines are given by the equations $2 l + 2 m - n = 0$ and $mn + n l + l m = 0$ , is :

(A)

\frac{π}{2}

(B)

π - cos^{- 1} (\frac{4}{9})

(C)

cos^{- 1} (\frac{8}{9})

(D)

\frac{π}{3}

JEE Main 2021 (25 February Shift 1)

LEVELJEE Main

Let $α$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l^{2} + m^{2} - n^{2} = 0$ . Then the value of $sin^{4} α + cos^{4} α$ is :

(A)

3/4

(B)

1/2

(C)

5/8

(D)

3/8

JEE Main 2022 (27 June Shift 1)

LEVELJEE Main

If two straight lines whose direction cosines are given by the relations $l + m - n = 0, 3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of $c$ is :

JEE Main 2024 (08 Apr Shift 1)

LEVELJEE Main

Let $P (x, y, z)$ be a point in the first octant, whose projection in the $x y$ -plane is the point $Q$ . Let $O P = γ$ ; the angle between $O Q$ and the positive $x$ -axis be $θ$ ; and the angle between $O P$ and the positive $z$ -axis be $ϕ$ , where $O$ is the origin. Then the distance of $P$ from the $x$ -axis is

(A)

γ 1 - sin^{2} ϕ cos^{2} θ

(B)

γ 1 - sin^{2} θ cos^{2} ϕ

(C)

γ 1 + cos^{2} ϕ sin^{2} θ

(D)

γ 1 + cos^{2} θ sin^{2} ϕ

JEE Main 2024 (27 Jan Shift 2)

LEVELJEE Main

The position vectors of the vertices $A, B$ and $C$ of a triangle are $2 \hat{i} - 3 \hat{j} + 3 \hat{k}$ , $2 \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $- \hat{i} + \hat{j} + 3 \hat{k}$ respectively. Let $l$ denotes the length of the angle bisector $A D$ of $∠ B A C$ where $D$ is on the line segment $B C$ , then $2 l^{2}$ equals :

Each of the angles $β$ and $γ$ that a given line makes with the positive $y$ - and $z$ -axes, respectively, is half of the angle that this line makes with the positive $x$ -axes. Then the sum of all possible values of the angle $β$ is

(A)

3 π /4

(B)

π

(C)

π /2

(D)

3 π /2

JEE Main 2025 (January)

LEVELJEE Main

Let $A (x, y, z)$ be a point in zy-plane, which is equidistant from three points (0, 3, 2), $(2, 0, 3)$ and $(0, 0, 1)$ . Let $B = (1, 4, - 1)$ and $C = (2, 0, - 2)$ . Then among the statements (S1): $△ A B C$ is an isosceles right angled triangle, and (S2): the area of $△ A B C$ is $\frac{9 2}{2}$

JEE Main 2026 (23 January Shift 1)

LEVELJEE Main

Let the direction cosines of two lines satisfy the equations : $4 l + m - n = 0$ and $2 mn + 10 n l + 3 l m = 0$ . Then the cosine of the acute angle between these lines is :

(A)

\frac{20}{3 38}

(B)

\frac{10}{3 38}

(C)

\frac{10}{7 38}

(D)

\frac{10}{38}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The angle between the lines whose direction cosines satisfy the equations l+m+n=0 and l2=m2+n2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a line makes an angle of π/4 with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2β=3sin2θ, then cos2θ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The angle between the straight lines, whose direction cosines are given by the equations 2l+2m−n=0 and mn+nl+lm=0, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α be the angle between the lines whose direction cosines satisfy the equations l+m−n=0 and l2+m2−n2=0. Then the value of sin4α+cos4α is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two straight lines whose direction cosines are given by the relations l+m−n=0,3l2+m2+cnl=0 are parallel, then the positive value of c is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the direction cosines of two lines satisfy the equations : 4l+m−n=0 and 2mn+10nl+3lm=0. Then the cosine of the acute angle between these lines is :

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^{2} = m^{2} + n^{2}$ is

If a line makes an angle of $π /4$ with the positive directions of each of $x$ -axis and $y$ -axis, then the angle that the line makes with the positive direction of the $z$ -axis is

A line makes the same angle $θ$ , with each of the $x$ and $z$ axis. If the angle $β$ , which it makes with $y$ -axis, is such that $sin^{2} β = 3 sin^{2} θ$ , then $cos^{2} θ$ equals

The angle between the straight lines, whose direction cosines are given by the equations $2 l + 2 m - n = 0$ and $mn + n l + l m = 0$ , is :

Let $α$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l^{2} + m^{2} - n^{2} = 0$ . Then the value of $sin^{4} α + cos^{4} α$ is :

If two straight lines whose direction cosines are given by the relations $l + m - n = 0, 3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of $c$ is :

Let the direction cosines of two lines satisfy the equations : $4 l + m - n = 0$ and $2 mn + 10 n l + 3 l m = 0$ . Then the cosine of the acute angle between these lines is :