The Cartesian equations of a line AB are 2 x - 1 2 = 4 - y 7 = z + 1 2 . Write the direction ratios of a line parallel to AB.

Given that, 2 x - 1 2 = 4 - y 7 = z + 1 2 In the cartesian equation the direction ratio can be obtained if the coefficient of x , y , z in the numerator is unity and positive for that. 2 x - 1 2 2 = y - 4 - 7 = z + 1 2 x - 1 2 1 = y - 4 - 7 = z + 1 2 Comparing with x - x 1 a = y - y 1 b = z - z 1 c Hence direction ratio a = 1 , b = - 7 , c = 2 Which is the required solution.

The Cartesian equations of a line AB are 2 x - 1 2 = 4 - y 7 = z + 1 2 . Write the direction ratios of a line parallel to AB.

Given that, 2 x - 1 2 = 4 - y 7 = z + 1 2 In the cartesian equation the direction ratio can be obtained if the coefficient of x , y , z in the numerator is unity and positive for that. 2 x - 1 2 2 = y - 4 - 7 = z + 1 2 x - 1 2 1 = y - 4 - 7 = z + 1 2 Comparing with x - x 1 a = y - y 1 b = z - z 1 c Hence direction ratio a = 1 , b = - 7 , c = 2 Which is the required solution.

The cartesian equation of a line AB is 2 x - 1 3 = y - 2 2 = z - 3 3 . Find the direction cosines of a line parallel to AB.

Given that, 2 x - 1 3 = y - 2 2 = z - 3 3 In the cartesian form the coefficient of x , y , z must be unity and positive to that. 2 x - 1 2 3 = y - 2 2 = z - 3 3 or x - 1 2 3 2 = y - 2 2 = z - 3 3 Comparing with x - x 1 a = y - y 1 b = z - z 1 c a = 3 2 , b = 2 , c = 3 or a : b : c = 3 2 : 2 : 3 Multiply by 2 , we get, a : b : c = 3 : 4 : 6 or a 1 = 3 , b 1 = 4 , c 1 = 6 Direction cosine of a line parallel to A B l = a a 2 + b 2 + c 2 , m = b a 2 + b 2 + c 2 , n = c a 2 + b 2 + c 2 Put all the values l = 3 ( 3 ) 2 + 4 2 + 6 2 , m = 4 ( 3 ) 2 + 4 2 + 6 2 , n = 6 ( 3 ) 2 + 4 2 + 6 2 ∴ l = 3 55 , m = 4 55 , n = 6 55

A line passes through the point with position vector 2 i ^ - j ^ + 4 k ^ and is in the direction of i ^ + j ^ - 2 k ^ . Find the equations of the line in vector and in cartesian form.

The vector equation of a straight line in space which passes through a fixed point A whose position vector is a → and is parallel to a given vector b → is given by r → = a → + λ b → , λ ∈ R . Here, a → = 2 i ^ - j ^ + 4 k ^ , b → = i ^ + j ^ - 2 k ^ Put a → , b → in r → Therefore, r → = 2 i ^ - j ^ + 4 k ^ + λ ( i ^ + j ^ - 2 k ^ ) Cartesian equation of the line passing through the point ( x , y , z ) and having direction ratios are a , b , c is given by x - x 1 a = y - y 1 b = z - z 1 c . . . 1 Passes through the point with position vector 2 i ^ - j ^ + 4 k ^ x 1 = 2 , y 1 = - 1 , z 1 = 4 And is in the direction of i ^ + j ^ - 2 k ^ Therefore, a = 1 , b = 1 , c = - 2 Put all the values in equation 1 ∴ x - 2 1 = y + 1 1 = z - 4 - 2

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 6

EASY

12th ICSE

IMPORTANT

Earn 100

The Cartesian equations of a line AB are $\frac{2 x - 1}{2} = \frac{4 - y}{7} = \frac{z + 1}{2}$ . Write the direction ratios of a line parallel to AB.

Important Questions on Three Dimensional Geometry

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 6

Write the direction cosines of a vector parallel to the line $\frac{4 - x}{2} = \frac{y + 3}{3} = \frac{z + 2}{6}$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 6

The cartesian equation of a line AB is $\frac{2 x - 1}{\sqrt{3}} = \frac{y - 2}{2} = \frac{z - 3}{3}$ . Find the direction cosines of a line parallel to AB.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 7

The cartesian equations of a line are

3 x - 3 = 2 y + 1 = 5 - 6 z

. Write the direction ratios of the line.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 7

Find the vector equation of the

x - axis .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 8

Find the vector and the cartesian equations of the line through the point $(5, 2, - 4)$ and which is parallel to the vector $3 \hat{i} + 2 \hat{j} - 8 \hat{k}$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 8

A line passes through the point with position vector

2 \hat{i} - \hat{j} + 4 \hat{k}

and is in the direction of

\hat{i} + \hat{j} - 2 \hat{k}

. Find the equations of the line in vector and in cartesian form.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 9

The cartesian equations of a line are $x = a y + b, z = c y + d$ . Find direction numbers of the line, also find its equation in vector form.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 10

Find the vector equation of the line passing through the point $A (2, - 1, 1)$ and parallel to the line joining the points $B (- 1, 4, 1) and C (1, 2, 2)$ . Also find the cartesian equation of the line.