Find the angle between the lines r → = ( 2 i ^ - 5 j ^ + k ^ ) + λ ( 3 i ^ + 2 j ^ + 6 k ^ ) and r → = ( 7 i ^ - 6 k ^ ) + μ ( i ^ + 2 j ^ + 2 k ^ )

Given lines are r → = ( 2 i ^ - 5 j ^ + k ^ ) + λ ( 3 i ^ + 2 j ^ + 6 k ^ ) and r → = ( 7 i ^ - 6 k ^ ) + μ ( i ^ + 2 j ^ + 2 k ^ ) Direction ratios of lines are 3 , 2 , 6 and 1 , 2 , 2 respectively The angle between the lines with direction ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 is given by cos θ = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 a 1 2 + b 1 2 + c 1 2 Hence angle between given lines is θ = cos - 1 3 ( 1 ) + 2 ( 2 ) + 6 ( 2 ) 3 2 + 2 2 + 6 2 1 2 + 2 2 + 2 2 ⇒ θ = cos - 1 3 + 4 + 12 9 + 4 + 36 1 + 4 + 4 = cos - 1 19 7 × 3 ⇒ θ = cos - 1 19 21 Hence, the angle between the lines r → = ( 2 i ^ - 5 j ^ + k ^ ) + λ ( 3 i ^ + 2 j ^ + 6 k ^ ) and r → = ( 7 i ^ - 6 k ^ ) + μ ( i ^ + 2 j ^ + 2 k ^ ) is cos - 1 19 21

Find the angle between the lines x + 3 3 = y - 1 5 = z + 3 4 and x + 1 1 = y - 4 1 = z - 5 2

Given lines are x + 3 3 = y - 1 5 = z + 3 4 and x + 1 1 = y - 4 1 = z - 5 2 Direction ratios of lines are 3 , 5 , 4 and 1 , 1 , 2 respectively The angle between the lines with direction ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 is given by cos θ = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 a 1 2 + b 1 2 + c 1 2 Hence angle between given lines is θ = cos - 1 3 ( 1 ) + 5 ( 1 ) + 4 ( 2 ) 3 2 + 5 2 + 4 2 1 2 + 1 2 + 2 2 θ = cos - 1 3 + 5 + 8 50 6 = cos - 1 16 10 3 = cos - 1 8 5 3 ⇒ θ = cos - 1 8 3 15 Hence the angle between the lines x + 3 3 = y - 1 5 = z + 3 4 and x + 1 1 = y - 4 1 = z - 5 2 is cos - 1 8 3 15

Show that the lines x - 5 7 = y + 2 - 5 = z 1 and x 1 = y 2 = z 3 are at right angles.

Given lines are x - 5 7 = y + 2 - 5 = z 1 and x 1 = y 2 = z 3 . Direction ratios of lines are 7 , - 5 , 1 and 1 , 2 , 3 respectively If two lines are perpendicular in 3 D then the sum of the product of the corresponding direction ratios is zero. For given lines we get 7 × 1 - 5 × 2 + 1 × 3 = 7 - 10 + 3 = 0 Hence proved that the given lines are perpendicular

What are the direction cosines of the vector ( 2 i ^ + j ^ - 2 k ^ ) ?

If a vector is l i → + m j → + n k → then direction cosines are given by l l 2 + m 2 + n 2 , m l 2 + m 2 + n 2 , n l 2 + m 2 + n 2 Direction cosines of the line with direction ratios 2 , 1 , - 2 is 2 2 2 + ( 1 ) 2 + ( - 2 ) 2 , 1 2 2 + ( 1 ) 2 + ( - 2 ) 2 , - 2 2 2 + ( 1 ) 2 + ( - 2 ) 2 = 2 4 + 1 + 4 , 1 4 + 1 + 4 , - 2 4 + 1 + 4 = 2 9 , 1 9 , - 2 9 = 2 3 , 1 3 , - 2 3 Direction cosines of the vector is = 2 3 , 1 3 , - 2 3

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 14

MEDIUM

12th CBSE

IMPORTANT

Earn 100

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

Important Questions on Straight Line in Space

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 15

Find the Cartesian equation of a line which passes through the point having position vector $(2 \hat{i} - \hat{j} + 4 \hat{k})$ and is in the direction of the vector $(\hat{i} + 2 \hat{j} - \hat{k})$

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 16

Find the angle between the lines $\vec{r} = (2 \hat{i} - 5 \hat{j} + \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})$ and $\vec{r} = (7 \hat{i} - 6 \hat{k}) + μ (\hat{i} + 2 \hat{j} + 2 \hat{k})$

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 17

Find the angle between the lines $\frac{x + 3}{3} = \frac{y - 1}{5} = \frac{z + 3}{4}$ and $\frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2}$

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 18

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 at right angles.

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 19

The direction ratios of a line are $2, 6, - 9$ . What are its direction cosines?

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 20

A line makes angles $90 °, 135 °$ and $45 °$ with the positive directions of x-axis, y-axis and z-axis respectively. What are the direction cosines of the line?

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 21

What are the direction cosines of the y-axis?

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 27 - Straight Line in Space>EXERCISE>Q 22

What are the direction cosines of the vector $(2 \hat{i} + \hat{j} - 2 \hat{k}) ?$