Find the angle between each of the following pairs of lines: r → = ( 3 i ^ - 4 j ^ + 2 k ^ ) + λ ( i ^ + 3 k ^ ) and r → = 5 i ^ + μ ( - i ^ + j ^ + k ^ )

Let L 1 → = ( 3 ı ^ - 4 ȷ ^ + 2 k ^ ) + λ ( ı ^ + 3 k ^ ) & L 2 → = ( 5 i ^ ) + μ ( - i ^ + j ^ + k ^ ) Direction ratios of L 1 = ( 1 , 0 , 3 ) Direction ratios of L 2 = ( - 1 , 1 , 1 ) I f ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second line then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 1 × ( - 1 ) + 0 × 1 + 3 × 1 1 2 + 0 2 + 3 2 1 2 + 1 2 + 1 2 = cos - 1 - 1 + 3 10 3 = cos - 1 2 30 Hence the angle between the given two lines = cos - 1 30 15

Find the angle between each of the following pairs of lines: r → = ( 3 i ^ - 4 j ^ + 2 k ^ ) + λ ( i ^ + 3 k ^ ) and r → = 5 i ^ + μ ( - i ^ + j ^ + k ^ )

Let L 1 → = ( 3 ı ^ - 4 ȷ ^ + 2 k ^ ) + λ ( ı ^ + 3 k ^ ) & L 2 → = ( 5 i ^ ) + μ ( - i ^ + j ^ + k ^ ) Direction ratios of L 1 = ( 1 , 0 , 3 ) Direction ratios of L 2 = ( - 1 , 1 , 1 ) I f ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second line then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 1 × ( - 1 ) + 0 × 1 + 3 × 1 1 2 + 0 2 + 3 2 1 2 + 1 2 + 1 2 = cos - 1 - 1 + 3 10 3 = cos - 1 2 30 Hence the angle between the given two lines = cos - 1 30 15

Find the angle between each of the following pairs of lines: r → = ( i ^ - 2 j ^ ) + λ ( 2 i ^ - 2 j ^ + k ^ ) and r → = 3 k ^ + μ ( i ^ + 2 j ^ - 2 k ^ )

Let L 1 → = ( 1 ^ - 2 ȷ ^ ) + λ ( 2 ı ^ - 2 ȷ ^ + k ^ ) & L 2 → = ( 3 k ^ ) + μ ( i ^ + 2 j ^ - 2 k ^ ) Direction ratios of L 1 = ( 2 , - 2 , 1 ) Direction ratios of L 2 = ( 1 , 2 , - 2 ) If ( a , b , c ) be the direction ratios of the first line and a ' , b ' , c ' be that of the second, then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 2 × 1 + ( - 2 ) × 2 + 1 × ( - 2 ) 2 2 + 2 2 + 1 2 1 2 + 2 2 + 2 2 = cos - 1 2 - 4 - 2 3 × 3 Hence the angle between the given two lines = cos - 1 - 4 9

Find the angle between each of the following pairs of lines: x - 1 1 = y - 4 1 = z - 5 2 and x + 3 3 = y - 2 5 = z + 5 4

Let L 1 → = x - 1 1 = y - 4 1 = z - 5 2 & L 2 → = x + 3 3 = y - 2 5 = z + 5 4 Direction ratios of L 1 = ( 1 , 1 , 2 ) Direction ratios of L 2 = ( 3 , 5 , 4 ) If ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second line then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 1 × 3 + 1 × 5 + 2 × 4 1 2 + 1 2 + 2 2 3 2 + 5 2 + 4 2 = cos - 1 3 + 5 + 8 6 × 50 Hence the angle between the given two lines = cos - 1 8 3 15

Find the angle between each of the following pairs of lines: x - 4 4 = y + 1 4 = z - 6 5 and x - 5 1 = 2 y + 5 - 2 = z - 3 1

Let L 1 → = x - 4 4 = y + 1 3 = z - 6 5 & L 2 → = x - 5 1 = y + 5 / 2 - 1 = z - 3 1 Direction ratios of L 1 = ( 4 , 3 , 5 ) Direction ratios of L 2 = ( 1 , - 1 , 1 ) If ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second, then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 4 × 1 + 3 × ( - 1 ) + 5 × 1 4 2 + 3 2 + 5 2 1 2 + 1 2 + 1 2 = cos - 1 4 - 3 + 5 5 2 × 3 = cos - 1 6 5 6 Hence the angle between the given two lines = cos - 1 2 6 15

Find the angle between each of the following pairs of lines: 3 - x - 2 = y + 5 1 = 1 - z 3 and x 3 = 1 - y - 2 = z + 2 - 1

Let L 1 → = x - 3 2 = y + 5 1 = z - 1 - 3 & L 2 → = x 3 = y - 1 2 = z + 2 - 1 Direction ratios of L 1 = ( 2 , 1 , - 3 ) Direction ratios of L 2 = ( 3 , 2 , - 1 ) If ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second, then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 2 × 3 + 1 × 2 + ( - 3 ) × ( - 1 ) 2 2 + 1 2 + 3 2 3 2 + 2 2 + 1 2 = cos - 1 6 + 2 + 3 14 × 14 Hence the angle between the given two lines = cos - 1 11 14

Find the angle between each of the following pairs of lines: x 1 = z - 1 , y = 0 and x 3 = y 4 = z 5

Let L 1 → = x 1 = y 0 = z - 1 & L 2 → = x 3 = y 4 = z 5 Direction ratios of L 1 = ( 1 , 0 , - 1 ) Direction ratios of L 2 = ( 3 , 4 , 5 ) If ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second, then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 1 × 3 + 0 × 4 + ( - 1 ) × 5 1 2 + 0 2 + 1 2 3 2 + 4 2 + 5 2 = cos - 1 3 - 5 5 2 × 2 Hence the angle between the given two lines = cos - 1 1 5

Find the angle between each of the following pairs of lines: 5 - x 3 = y + 3 - 2 , z = 5 and x - 1 1 = 1 - y 3 = z - 5 2

Let L 1 → = x - 5 - 3 = y + 3 - 2 = z - 5 0 & L 2 → = x - 1 1 = y - 1 - 3 = z - 5 2 Direction ratios of L 1 = ( - 3 , - 2 , 0 ) Direction ratios of L 2 = ( 1 , - 3 , 2 ) I f ( a , b , c ) be the direction ratios of the first line and ( a ’ , b ’ , c ’ ) be that of the second, then the angle between these pair of lines is given by cos - 1 a × a ' + b × b ' + c × c ' a 2 + b 2 + c 2 × a ' 2 + b ' 2 + c ' 2 The angle between the lines = cos - 1 ( - 3 ) × 1 + ( - 2 ) × ( - 3 ) + 0 × 2 3 2 + 2 2 + 0 2 1 2 + 3 2 + 2 2 = cos - 1 - 3 + 6 13 × 14 Hence the angle between the given two lines = cos - 1 3 182

Show that the lines x - 3 2 = y + 1 - 3 = z - 2 4 and x + 2 2 = y - 4 4 = z + 5 2 are perpendicular to each other.

Let L 1 → = x - 3 2 = y + 1 - 3 = z - 2 4 & L 2 → = x + 2 2 = y - 4 4 = z + 5 2 Direction ratios of L 1 = ( 2 , - 3 , 4 ) Direction ratios of L 2 = ( 2 , 4 , 2 ) If two lines are perpendicular then the sum of the product of respective direction ratios is zero ⇒ 2 × 2 - 3 × 4 + 4 × 2 = 4 - 12 + 8 = 0 Hence the lines are perpendicular to each other

If the lines x - 1 - 3 = y - 2 2 λ = z - 3 2 and x - 1 3 λ = y - 1 1 = 6 - z 5 are perpendicular to each other then find the value of λ .

Let L 1 → = x - 1 - 3 = y - 2 2 λ = z - 3 2 & L 2 → = x - 1 3 λ = y - 1 1 = z - 6 - 5 Direction ratios of L 1 = ( - 3 , 2 λ , 2 ) Direction ratios of L 2 = ( 3 λ , 1 , - 5 ) If two lines are perpendicular then the sum of the product of respective direction ratios is zero ⇒ - 9 λ + 2 λ - 10 = 0 ⇒ λ = - 10 7

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 2

EASY

12th CBSE

IMPORTANT

Earn 100

Find the angle between each of the following pairs of lines:

$\vec{r} = (3 \hat{i} - 4 \hat{j} + 2 \hat{k}) + λ (\hat{i} + 3 \hat{k}) and \vec{r} = 5 \hat{i} + μ (- \hat{i} + \hat{j} + \hat{k})$

Important Questions on Straight Line in Space

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 3

Find the angle between each of the following pairs of lines:

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

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 4

Find the angle between each of the following pairs of lines:

$\frac{x - 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2} and \frac{x + 3}{3} = \frac{y - 2}{5} = \frac{z + 5}{4}$

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 5

Find the angle between each of the following pairs of lines:

$\frac{x - 4}{4} = \frac{y + 1}{4} = \frac{z - 6}{5} and \frac{x - 5}{1} = \frac{2 y + 5}{- 2} = \frac{z - 3}{1}$

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 6

Find the angle between each of the following pairs of lines:

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

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 7

Find the angle between each of the following pairs of lines:

$\frac{x}{1} = \frac{z}{- 1}, y = 0 and \frac{x}{3} = \frac{y}{4} = \frac{z}{5}$

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 8

Find the angle between each of the following pairs of lines:

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

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 9

Show that the lines $\frac{x - 3}{2} = \frac{y + 1}{- 3} = \frac{z - 2}{4} and \frac{x + 2}{2} = \frac{y - 4}{4} = \frac{z + 5}{2}$ are perpendicular to each other.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 10

If the lines $\frac{x - 1}{- 3} = \frac{y - 2}{2 λ} = \frac{z - 3}{2} and \frac{x - 1}{3 λ} = \frac{y - 1}{1} = \frac{6 - z}{5}$ are perpendicular to each other then find the value of $λ .$