If a Line Makes Angles α, β and γ with the Axes Respectively, Then Cos 2 α + Cos 2 β + Cos 2 γ = (A) −2 (B) −1 (C) 1 (D) 2 - Mathematics

प्रश्न

If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =

विकल्प

−2
−1
1
2

MCQ

उत्तर

−1

If a line makes angles α, β and γ with the axes, then

$\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

We have ,

$\cos 2 α + \cos 2 β + \cos 2 γ = 2 \cos^{2} α - 1 + 2 \cos^{2} β - 1 + 2 \cos^{2} γ - 1 [∵ \cos 2 θ = 2 \cos^{2} θ - 1]$

$= 2 (\cos^{2} α + \cos^{2} β + \cos^{2} γ) - 3 [From (1)]$

$= 2 (1) - 3$

$= - 1$

shaalaa.com

Equation of a Line in Space

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 8 Q 7 Q 9

अध्याय 28: Straight Line in Space - MCQ [पृष्ठ ४३]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 28 Straight Line in Space
MCQ | Q 8 | पृष्ठ ४३

वीडियो ट्यूटोरियलVIEW ALL [4]

view
वीडियो ट्यूटोरियल For All Subjects
Equation of a Line in Space

video tutorial00:21:23
Equation of a Line in Space

video tutorial00:03:54
Equation of a Line in Space

video tutorial00:31:17
Equation of a Line in Space

video tutorial00:15:14

संबंधित प्रश्न

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines.

$\vec{r} = (8 \hat{i} - 19 \hat{j} + 10 \hat{k}) + λ (3 \hat{i} - 16 \hat{j} + 7 \hat{k}) and \vec{r} = (15 \hat{i} + 29 \hat{j} + 5 \hat{k}) + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})$

Find the value of p, so that the lines $l_{1} : \frac{1 - x}{3} = \frac{7 y - 14}{p} = \frac{z - 3}{2} and l_{2} = \frac{7 - 7 x}{3} p = \frac{y - 5}{1} = \frac{6 - z}{5}$ are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

Let $A (\bar{a})$ and $B (\bar{b})$ be any two points in the space and $R (\bar{r})$ be a point on the line segment AB dividing it internally in the ratio m : n, then prove that $\bar{r} = \frac{m \bar{b} + n \bar{a}}{m + n}$ . Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

The Cartesian equation of a line is $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$ Write its vector form.

Show that the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each other.

Find the equation of a line parallel to x-axis and passing through the origin.

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).

Find the vector equation of the line passing through the points (−1, 0, 2) and (3, 4, 6).

Find the angle between the following pair of line:

$\vec{r} = λ (\hat{i} + \hat{j} + 2 \hat{k}) and \vec{r} = 2 \hat{j} + μ {(\sqrt{3} - 1) \hat{i} - (\sqrt{3} + 1) \hat{j} + 4 \hat{k}}$

Find the angle between the following pair of line:

$\frac{5 - x}{- 2} = \frac{y + 3}{1} = \frac{1 - z}{3} and \frac{x}{3} = \frac{1 - y}{- 2} = \frac{z + 5}{- 1}$

Find the angle between the following pair of line:

$\frac{x - 2}{3} = \frac{y + 3}{- 2}, z = 5 and \frac{x + 1}{1} = \frac{2 y - 3}{3} = \frac{z - 5}{2}$

Find the equations of the line passing through the point (2, 1, 3) and perpendicular to the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} and \frac{x}{- 3} = \frac{y}{2} = \frac{z}{5}$

Find the equation of the line passing through the point $\hat{i} + \hat{j} - 3 \hat{k}$ and perpendicular to the lines $\vec{r} = \hat{i} + λ (2 \hat{i} + \hat{j} - 3 \hat{k}) and \vec{r} = (2 \hat{i} + \hat{j} - \hat{k}) + μ (\hat{i} + \hat{j} + \hat{k}) .$

Find the value of λ so that the following lines are perpendicular to each other. $\frac{x - 5}{5 λ + 2} = \frac{2 - y}{5} = \frac{1 - z}{- 1}, \frac{x}{1} = \frac{2 y + 1}{4 λ} = \frac{1 - z}{- 3}$

Find the direction cosines of the line

$\frac{x + 2}{2} = \frac{2 y - 7}{6} = \frac{5 - z}{6}$ Also, find the vector equation of the line through the point A(−1, 2, 3) and parallel to the given line.

Show that the lines $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} and \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ intersect and find their point of intersection.

Determine whether the following pair of lines intersect or not:

$\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} a n d \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}$

Show that the lines $\vec{r} = 3 \hat{i} + 2 \hat{j} - 4 \hat{k} + λ (\hat{i} + 2 \hat{j} + 2 \hat{k}) and \vec{r} = 5 \hat{i} - 2 \hat{j} + μ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})$ are intersecting. Hence, find their point of intersection.

Find the equation of the perpendicular drawn from the point P (2, 4, −1) to the line $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{- 9} .$ Also, write down the coordinates of the foot of the perpendicular from P.

Find the foot of the perpendicular from (1, 2, −3) to the line $\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1} .$

Find the shortest distance between the following pairs of lines whose vector equations are: $\vec{r} = (8 + 3 λ) \hat{i} - (9 + 16 λ) \hat{j} + (10 + 7 λ) \hat{k}$ $\vec{r} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})$

By computing the shortest distance determine whether the following pairs of lines intersect or not: $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + λ (3 \hat{i} - \hat{j}) and \vec{r} = (4 \hat{i} - \hat{k}) + μ (2 \hat{i} + 3 \hat{k})$

Write the cartesian and vector equations of Z-axis.

Write the angle between the lines 2x = 3y = −z and 6x = −y = −4z.

Find the Cartesian equations of the line which passes through the point (−2, 4 , −5) and is parallel to the line $\frac{x + 3}{3} = \frac{4 - y}{5} = \frac{z + 8}{6} .$

The angle between the straight lines $\frac{x + 1}{2} = \frac{y - 2}{5} = \frac{z + 3}{4} a n d \frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 3}{- 3}$ is

The direction ratios of the line perpendicular to the lines $\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} and, \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}$ are proportional to

The angle between the lines

\frac{x - 1}{1} = \frac{y - 1}{1} = \frac{z - 1}{2} and, \frac{x - 1}{- \sqrt{3} - 1} = \frac{y - 1}{\sqrt{3} - 1} = \frac{z - 1}{4}

If the direction ratios of a line are proportional to 1, −3, 2, then its direction cosines are

The straight line $\frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}$ is

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}$ .

Find the value of p for which the following lines are perpendicular :

$\frac{1 - x}{3} = \frac{2 y - 14}{2 p} = \frac{z - 3}{2}; \frac{1 - x}{3 p} = \frac{y - 5}{1} = \frac{6 - z}{5}$

Find the value of λ, so that the lines $\frac{1 - x}{3} = \frac{7 y - 14}{λ} = \frac{z - 3}{2} and \frac{7 - 7 x}{3 λ} = \frac{y - 5}{1} = \frac{6 - z}{5}$ are at right angles. Also, find whether the lines are intersecting or not.

Choose correct alternatives:

The difference between the slopes of the lines represented by 3x² - 4xy + y² = 0 is 2

Find the vector equation of a line passing through a point with position vector $2 \hat{i} - \hat{j} + \hat{k}$ and parallel to the line joining the points $- \hat{i} + 4 \hat{j} + \hat{k}$ and $- \hat{i} + 2 \hat{j} + 2 \hat{k}$ .

A line passes through the point (2, – 1, 3) and is perpendicular to the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + λ (2 \hat{i} - 2 \hat{j} + \hat{k})$ and $\vec{r} = (2 \hat{i} - \hat{j} - 3 \hat{k}) + μ (\hat{i} + 2 \hat{j} + 2 \hat{k})$ obtain its equation.

If a Line Makes Angles α, β and γ with the Axes Respectively, Then Cos 2 α + Cos 2 β + Cos 2 γ = (A) −2 (B) −1 (C) 1 (D) 2 - Mathematics

Advertisements

Advertisements

प्रश्न

विकल्प

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [4]

संबंधित प्रश्न

Select a course

If a Line Makes Angles α, β and γ with the Axes Respectively, Then Cos 2 α + Cos 2 β + Cos 2 γ = (A) −2 (B) −1 (C) 1 (D) 2 - Mathematics

Advertisements

Advertisements

प्रश्न

विकल्प

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [4]

संबंधित प्रश्न

Select a course

उत्तर