Show that the following straight lines are coplanar x + 3 2 = y + 5 3 = z - 7 - 3 and x + 1 4 = y + 1 5 = z + 1 - 1

We know that the lines x - x 1 l 1 = y - y 1 m 1 = z - z 1 n 1 and x - x 2 l 2 = y - y 2 m 2 = z - z 2 n 2 are coplanar, then x 2 - x 1 y 2 - y 1 z 2 - z 1 l 1 m 1 n 1 l 2 m 2 n 2 = 0 The given lines are: x + 3 2 = y + 5 3 = z - 7 - 3 and x + 1 4 = y + 1 5 = z + 1 - 1 Consider, - 1 - - 3 - 1 - - 5 - 1 - 7 2 3 - 3 4 5 - 1 = 2 4 - 8 2 3 - 3 4 5 - 1 = 2 - 3 + 15 - 4 - 2 + 12 - 8 10 - 12 = 24 - 40 + 16 = 0 Therefore, the given lines are coplanar.

Show that the following straight lines are coplanar x + 3 2 = y + 5 3 = z - 7 - 3 and x + 1 4 = y + 1 5 = z + 1 - 1

We know that the lines x - x 1 l 1 = y - y 1 m 1 = z - z 1 n 1 and x - x 2 l 2 = y - y 2 m 2 = z - z 2 n 2 are coplanar, then x 2 - x 1 y 2 - y 1 z 2 - z 1 l 1 m 1 n 1 l 2 m 2 n 2 = 0 The given lines are: x + 3 2 = y + 5 3 = z - 7 - 3 and x + 1 4 = y + 1 5 = z + 1 - 1 Consider, - 1 - - 3 - 1 - - 5 - 1 - 7 2 3 - 3 4 5 - 1 = 2 4 - 8 2 3 - 3 4 5 - 1 = 2 - 3 + 15 - 4 - 2 + 12 - 8 10 - 12 = 24 - 40 + 16 = 0 Therefore, the given lines are coplanar.

Show that the following straight lines are coplanar r → = i ^ + j ^ - k ^ + t ( 3 i ^ - j ^ ) and r → = ( 4 i ^ - k ^ ) + t ( 2 i ^ + 3 k ^ )

We know that the lines r → = a 1 → + λ b 1 → and r → = a 2 → + μ b 2 → are coplanar, if a 2 → - a 1 → b 1 → b 2 → = 0 . Here, a 1 → = i ^ + j ^ - k ^ , b 1 → = 3 i ^ - j ^ , a 2 → = 4 i ^ + 0 j ^ - k ^ and b 2 → = 2 i ^ + 0 j ^ + 3 k ^ . Now, a 2 → - a 1 → = 4 i ^ + 0 j ^ - k ^ - i ^ + j ^ - k ^ = 3 i ^ - j ^ So, a 2 → - a 1 → b 1 → b 2 → = 3 - 1 0 3 - 1 0 2 0 3 = 3 - 3 - 0 + 1 9 - 0 + 0 0 + 2 = - 9 + 9 = 0 Hence, proved that the given lines are coplanar.

Test whether the following pair of lines intersect or not. x - 3 2 = y + 15 - 7 = z - 9 5 , x + 1 2 = y - 1 1 = z - 9 - 3

We know that if the lines x - x 1 l 1 = y - y 1 m 1 = z - z 1 n 1 and x - x 2 l 2 = y - y 2 m 2 = z - z 2 n 2 are intersecting then they are coplanar and directions ratios of both lines are not proportional, so x 2 - x 1 y 2 - y 1 z 2 - z 1 l 1 m 1 n 1 l 2 m 2 n 2 = 0 The given lines are: x - 3 2 = y + 15 - 7 = z - 9 5 , x + 1 2 = y - 1 1 = z - 9 - 3 Consider, - 1 - 3 1 - - 15 9 - 9 2 - 7 5 2 1 - 3 = - 4 16 0 2 - 7 5 2 1 - 3 = - 4 21 - 5 - 16 - 6 - 10 + 0 2 + 14 = - 4 16 - 16 - 16 = - 64 + 256 ≠ 0 So, the given lines are not coplanar. Therefore, they are not intersecting.

Test whether the following pair of lines intersect or not. x - 1 2 = y - 2 3 = z - 3 4 , x - 2 3 = y - 3 4 = z - 4 5

We know that if the lines x - x 1 l 1 = y - y 1 m 1 = z - z 1 n 1 and x - x 2 l 2 = y - y 2 m 2 = z - z 2 n 2 are intersecting then they are coplanar and directions ratios of both lines are not proportional, so x 2 - x 1 y 2 - y 1 z 2 - z 1 l 1 m 1 n 1 l 2 m 2 n 2 = 0 The given lines are: x - 1 2 = y - 2 3 = z - 3 4 , x - 2 3 = y - 3 4 = z - 4 5 Consider, 2 - 1 3 - 2 4 - 3 2 3 4 3 4 5 = 1 1 1 2 3 4 3 4 5 = 1 15 - 16 - 1 10 - 12 + 1 8 - 9 = - 1 + 2 - 1 = 0 So, the given lines are coplanar. The direction ratios of two lines are not proportional. Therefore, they are not intersecting.

Test whether the following pair of lines intersect or not. r → = - i ^ - 3 j ^ - 5 k ^ + t ( 3 i ^ + 5 j ^ + 7 k ^ ) , r → = 2 i ^ + 4 j ^ + 6 k ^ + t ( i ^ + 3 j ^ + 5 k ^ )

The lines r → = a 1 → + λ b 1 → and r → = a 2 → + μ b 2 → intersects if they are coplanar and vectors b 1 → , b 2 → are non collinear, so a 2 → - a 1 → b 1 → b 2 → = 0 . Here, a 1 → = - i ^ - 3 j ^ - 5 k ^ , b 1 → = 3 i ^ + 5 j ^ + 7 k ^ , a 2 → = 2 i ^ + 4 j ^ + 6 k ^ and b 2 → = i ^ + 3 j ^ + 5 k ^ . Now, a 2 → - a 1 → = 2 i ^ + 4 j ^ + 6 k ^ - - i ^ - 3 j ^ - 5 k ^ = 3 i ^ + 7 j ^ + 11 k ^ So, a 2 → - a 1 → b 1 → b 2 → = 3 7 11 3 5 7 1 3 5 = 3 25 - 21 - 7 15 - 7 + 11 9 - 5 = 3 4 - 7 8 + 11 4 = 12 - 56 + 44 = 0 So, the given lines are coplanar. The vectors b 1 → , b 2 → are non collinear. Therefore, the given lines intersect each other.

E M B I B E

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 13

EASY

12th West Bengal Board

IMPORTANT

Earn 100

Show that the following straight lines are coplanar
$\frac{x + 3}{2} = \frac{y + 5}{3} = \frac{z - 7}{- 3}$ and $\frac{x + 1}{4} = \frac{y + 1}{5} = \frac{z + 1}{- 1}$

Important Questions on Straight Line in Three Dimensions

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 13

Show that the following straight lines are coplanar

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

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 14

For what value of

k

, the line

\frac{x - 1}{2} = \frac{y - 4}{1} = \frac{z - 5}{2}

intersects the line

\frac{x - 2}{- 1} = \frac{y - 8}{k} = \frac{z - 11}{4}

?

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 15

Test whether the following pair of lines intersect or not.

$\frac{x - 3}{2} = \frac{y + 15}{- 7} = \frac{z - 9}{5}, \frac{x + 1}{2} = \frac{y - 1}{1} = \frac{z - 9}{- 3}$

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 15

Test whether the following pair of lines intersect or not.

\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}, \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 24 - Straight Line in Three Dimensions>EXERCISE>Q 15

Test whether the following pair of lines intersect or not.

$\vec{r} = - \hat{i} - 3 \hat{j} - 5 \hat{k} + t (3 \hat{i} + 5 \hat{j} + 7 \hat{k}), \vec{r} = 2 \hat{i} + 4 \hat{j} + 6 \hat{k} + t (\hat{i} + 3 \hat{j} + 5 \hat{k})$