Geometry

3D Geometry

Medium

core

The direction cosines of a line which makes equal angles with co-ordinate axes are:

$\pm(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$

$\pm(1, 1, 1)$

$\pm(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$

$\pm(\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{-1}{\sqrt{3}})$

✅ Correct Option: 1

Direction cosines are the cosines of the angles that a line makes with the x, y, and z axes. If a line makes angles α, β, and γ with the three axes, the direction cosines are:

$l = \cos α$ (angle with x-axis)

$m = \cos β$ (angle with y-axis)

$n = \cos γ$ (angle with z-axis)

These satisfy the relation: $l^2 + m^2 + n^2 = 1$

The line makes equal angles with all three coordinate axes, so:

$α = β = γ = θ$

The direction cosines become:

$l = \cos θ$

$m = \cos θ$

$n = \cos θ$

Using $l^2 + m^2 + n^2 = 1$ :

$(\cos θ)^2 + (\cos θ)^2 + (\cos θ)^2 = 1$

$3 \cos^2 θ = 1$

$\cos^2 θ = \dfrac{1}{3}$

$\cos θ = ±\dfrac{1}{\sqrt{3}}$

Since $l = m = n = \cos θ$ :

Direction cosines $= \left(±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}\right)$

The $±$ sign accounts for the line pointing in two opposite directions.

2025: 21 May Shift 2

The angle between the line $2x = 3y = z$ and $x$ - axis is:

2025: 2 June Shift 1

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

2025: 3 June Shift 1

If the lines $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ and $x = \frac{y + 1/2}{2\lambda} = \frac{z - 1}{3}$ are perpendicular, then the value of $\lambda$ is equal to

2025: 26 May Shift 2

A line passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction of the vector $\hat{i} + \hat{j} - 2\hat{k}$ . The equation of the line in Cartesian form is:

2025: 21 May Shift 1

The acute angle between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\frac{x-1}{1} = \frac{y+1}{-3} = \frac{z-2}{2}$ is

2022: 25 May Shift 1

A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector $3\hat{i} + 2\hat{j} - 2\hat{k}$ is $\frac{x-1}{3} = \frac{y-2}{2} = \frac{y-3}{-2}$ .

B. Equation of line passing through (1, 2, 3) and parallel to the line given by $\frac{x+3}{3} = \frac{4-y}{5} = \frac{z+8}{6}$ is $\frac{x-1}{3} = \frac{y-2}{5} = \frac{z+3}{6}$ .

C. Equation of line passing through the origin and (5, -2, 3) is $\frac{x}{5} = \frac{y}{-2} = \frac{z}{3}$ .

D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is $2(x-1)+3(y-2)-1(z-3) = 0$ .

E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is $2x + 3y + 4z = 1$ .

Choose the correct answer from the options given below:

The direction cosines of a line which makes equal angles with co-ordinate axes are:

Solution

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