Find the angle between the following pair of line: r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 2 k ^ ) and r → = i ^ - j ^ + 2 k ^ - μ ( 2 i ^ + 4 j ^ - 4 k ^ )

r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 2 k ^ ) and r → = i ^ - j ^ + 2 k ^ - μ ( 2 i ^ + 4 j ^ - 4 k ^ ) Let b → 1 and b → 2 be vectors parallel to the given lines. Now, b 1 → = i ^ + 2 j ^ - 2 k ^ and b 2 → = 2 i ^ + 4 j ^ - 4 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( i ^ + 2 j ^ - 2 k ^ ) · ( 2 i ^ + 4 j ^ - 4 k ^ ) 1 2 + 2 2 + ( - 2 ) 2 2 2 + 4 2 + ( - 4 ) 2 = 2 + 8 + 8 3 × 6 = 1 ⇒ θ = 0 °

Find the angle between the following pair of line: r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 2 k ^ ) and r → = i ^ - j ^ + 2 k ^ - μ ( 2 i ^ + 4 j ^ - 4 k ^ )

r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 2 k ^ ) and r → = i ^ - j ^ + 2 k ^ - μ ( 2 i ^ + 4 j ^ - 4 k ^ ) Let b → 1 and b → 2 be vectors parallel to the given lines. Now, b 1 → = i ^ + 2 j ^ - 2 k ^ and b 2 → = 2 i ^ + 4 j ^ - 4 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( i ^ + 2 j ^ - 2 k ^ ) · ( 2 i ^ + 4 j ^ - 4 k ^ ) 1 2 + 2 2 + ( - 2 ) 2 2 2 + 4 2 + ( - 4 ) 2 = 2 + 8 + 8 3 × 6 = 1 ⇒ θ = 0 °

Find the angle between the lines r → = 3 i ^ + 2 j ^ - 4 k ^ + λ i ^ + 2 j ^ + 2 k ^ and r → = 5 j ^ - 2 k ^ + μ 3 i ^ + 2 j ^ + 6 k ^

r → = ( 3 i ^ + 2 j ^ - 4 k ^ ) + λ ( i ^ + 2 j ^ + 2 k ^ ) and r → = ( 5 j ^ - 2 k ^ ) + μ ( 3 i ^ + 2 j ^ + 6 k ^ ) Let b 1 → and b 2 → be vector parallel to the given line. Now, b 1 → = i ^ + 2 j ^ + 2 k ^ and b 2 → = 3 i ^ + 2 j ^ + 6 k ^ If θ is the angle between the given line, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( i ^ + 2 j ^ + 2 k ^ ) · ( 3 i ^ + 2 j ^ + 6 k ^ ) 1 2 + 2 2 + 2 2 3 2 + 2 2 + 6 2 = 3 + 4 + 12 3 × 7 = 19 21 ⇒ θ = cos - 1 19 21

Find the angle between the following pair of line: r → = λ i ^ + j ^ + 2 k ^ and r → = 2 j ^ + μ 3 - 1 i ^ - 3 + 1 j ^ + 4 k ^

r → = λ ( i ^ + j ^ + 2 k ^ ) and r → = 2 j ^ + μ { ( 3 - 1 ) i ^ - ( 3 + 1 ) j ^ + 4 k ^ } Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = 2 i ^ + 3 j ^ - 3 k ^ and b 2 → = ( 3 - 1 ) i ^ - ( 3 + 1 ) j ^ + 4 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( i ^ + j ^ + 2 k ^ ) · ( ( 3 - 1 ) i ^ - ( 3 + 1 ) j ^ + 4 k ^ ) 1 2 + 1 2 + 2 2 ( 3 - 1 ) 2 + ( 3 + 1 ) 2 + 4 2 = ( 3 - 1 ) - ( 3 + 1 ) + 8 6 24 = 6 12 = 1 2 ⇒ θ = π 3

Find the angle between the following pair of line: x + 4 3 = y - 1 5 = z + 3 4 and x + 1 1 = y - 4 1 = z - 5 2

x + 4 3 = y - 1 5 = z + 3 4 and x + 1 1 = y - 4 1 = z - 5 2 Let b 1 → and b 2 → be vectors parallel to the given line. b 1 → = 3 i ^ + 5 j ^ + 4 k ^ and b 2 → = i ^ + j ^ + 2 k ^ If θ is the angle between the given line, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( 3 i ^ + 5 j ^ + 4 k ^ ) · ( i ^ + j ^ + 2 k ^ ) 3 2 + 5 2 + 4 2 1 2 + 1 2 + 2 2 = 3 + 5 + 8 10 3 = 8 5 3 ⇒ θ = cos - 1 8 5 3

Find the angle between the following pair of line: x - 1 2 = y - 2 3 = z - 3 - 3 and x + 3 - 1 = y - 5 8 = z - 1 4

x - 1 2 = y - 2 3 = z - 3 - 3 and x + 3 - 1 = y - 5 8 = z - 1 4 Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = 2 i ^ + 3 j ^ - 3 k ^ and b 2 → = - i ^ + 8 j ^ + 4 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( 2 i ^ + 3 j ^ - 3 k ^ ) · ( - i ^ + 8 j ^ + 4 k ^ ) 2 2 + 3 2 + ( - 3 ) 2 ( - 1 ) 2 + 8 2 + 4 2 = - 2 + 24 - 12 9 22 = 10 9 22 ⇒ θ = cos - 1 10 9 22

Find the angle between the following pair of line: 5 - x - 2 = y + 3 1 = 1 - z 3 and x 3 = 1 - y - 2 = z + 5 - 1

5 - x - 2 = y + 3 1 = 1 - z 3 and x 3 = 1 - y - 2 = z + 5 - 1 The equation of the given line can be re-written as x - 5 2 = y + 3 1 = z - 1 - 3 and x 3 = y - 1 2 = z + 5 - 1 Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = 2 i ^ + j ^ - 3 k ^ and b 2 → = 3 i ^ + 2 j ^ - k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( 2 i ^ + j ^ - 3 k ^ ) · ( 3 i ^ + 2 j ^ - k ^ ) 2 2 + 1 2 + ( - 3 ) 2 3 2 + 2 2 + ( - 1 ) 2 = 6 + 2 + 3 14 14 = 11 14 ⇒ θ = cos - 1 11 14

Find the angle between the following pair of line: x - 2 3 = y + 3 - 2 , z = 5 and x + 1 1 = 2 y - 3 3 = z - 5 2

x - 2 3 = y + 3 - 2 , z = 5 and x + 1 1 = 2 y - 3 3 = z - 5 2 The equations of the given lines can be re-written as x - 2 3 = y + 3 - 2 = z - 5 0 and x + 1 1 = y - 3 2 3 2 = z - 5 2 Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = 3 i ^ - 2 j ^ + 0 k ^ and b 2 → = i ^ + 3 2 j ^ + 2 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( 3 i ^ - 2 j ^ + 0 k ^ ) · i ^ + 3 2 j ^ + 2 k ^ 3 2 + ( - 2 ) 2 + 0 2 1 2 + 3 2 2 + 2 2 = 3 - 3 + 0 13 29 4 = 0 ⇒ θ = π 2

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

x - 5 1 = 2 y + 6 - 2 = z - 3 1 and x - 2 3 = y + 1 4 = z - 6 5 The equations of the given lines can be re-written as x - 5 1 = y + 3 - 1 = z - 3 1 and x - 2 3 = y + 1 4 = z - 6 5 Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = i ^ - j ^ + k ^ and b 2 → = 3 i ^ + 4 j ^ + 5 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( i ^ - j ^ + k ^ ) · ( 3 i ^ + 4 j ^ + 5 k ^ ) 1 2 + ( - 1 ) 2 + 1 2 3 2 + 4 2 + 5 2 = 3 - 4 + 5 3 50 = 4 5 6 ⇒ θ = cos - 1 4 5 6

Find the angle between the following pair of line: - x + 2 - 2 = y - 1 7 = z + 3 - 3 and x + 2 - 1 = 2 y - 8 4 = z - 5 4

- x + 2 - 2 = y - 1 7 = z + 3 - 3 and x + 2 - 1 = 2 y - 8 4 = z - 5 4 The equations of the given lines can be re-written as x - 2 2 = y - 1 7 = z + 3 - 3 and x + 2 - 1 = y - 4 2 = z - 5 4 Let b 1 → and b 2 → be vectors parallel to the given lines. Now, b 1 → = 2 i ^ + 7 j ^ - 3 k ^ and b 2 → = - 1 i ^ + 2 j ^ + 4 k ^ If θ is the angle between the given lines, then cos θ = b 1 → · b 2 → b 1 → b 2 → = ( 2 i ^ + 7 j ^ - 3 k ^ ) · ( - 1 i ^ + 2 j ^ + 4 k ^ ) 2 2 + 7 2 + ( - 3 ) 2 ( - 1 ) 2 + 2 2 + 4 2 = - 2 + 14 - 12 62 21 = 0 ⇒ θ = π 2

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 9 - Straight Line in Space>EXERCISE>Q 8

MEDIUM

12th CBSE

IMPORTANT

Earn 100

Find the angle between the following pair of line:

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

Important Questions on Straight Line in Space

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 9 - Straight Line in Space>EXERCISE>Q 8

Find the angle between the lines $\vec{r} = (3 \hat{i} + 2 \hat{j} - 4 \hat{k}) + λ (\hat{i} + 2 \hat{j} + 2 \hat{k}) and \vec{r} = (5 \hat{j} - 2 \hat{k}) + μ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})$

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 9 - Straight Line in Space>EXERCISE>Q 8

Find the angle between the following pair of line: $\vec{r} = λ (\hat{i} + \hat{j} + 2 \hat{k}) and \vec{r} = 2 \hat{j} + μ \{(\sqrt{3} - 1) \hat{i} - (\sqrt{3} + 1) \hat{j} + 4 \hat{k}\}$