Find the angle between the following pair of lines:
$\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4}$
And check whether the lines are parallel or perpendicular.

Solution

$Let {\vec{b}}_{1} and {\vec{b}}_{2} be the two vector parallel to the pair to lines, \frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4}, respectively . Now, \frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} \Rightarrow \frac{x - 2}{2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4} \Rightarrow \frac{x + 2}{- 1} = \frac{y - 4}{2} = \frac{z - 5}{4}$

$∴ {\vec{b}}_{1} = 2 \hat{i} + 7 \hat{j} - 3 \hat{k} and {\vec{b}}_{2} = - \hat{i} + 2 \hat{j} + 4 \hat{k} |{\vec{b}}_{1}| = \sqrt{{(2)}^{2} + {(7)}^{2} + {(- 3)}^{2}} = \sqrt{62} |{\vec{b}}_{2}| = \sqrt{{(- 1)}^{2} + {(2)}^{2} + {(4)}^{2}} = \sqrt{21} {\vec{b}}_{1} . {\vec{b}}_{2} = (2 \hat{i} + 7 \hat{j} - 3 \hat{k}) . (- \hat{i} + 2 \hat{j} + 4 \hat{k}) = 2 (- 1) + 7 x 2 + (- 3) . 4$

= - 2 + 14 - 12

= 0

The angle $θ$ between the given pair of lines is given by the relation,

$\cos θ = |\frac{\vec{b_{1}} . \vec{b_{2}}}{|{\vec{b}}_{1}| |\vec{b_{2}}|}| \Rightarrow \cos θ = \frac{0}{\sqrt{62} x \sqrt{21}} = 0 \Rightarrow θ = \cos^{- 1} (0) = \frac{π}{2}$

Thus, the given lines are perpendicular to each other and the angle

between them is $90^{0}$ .

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the angle between the following pair of lines:
$\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4}$
And check whether the lines are parallel or perpendicular.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the angle between the following pair of lines: - x + 2- 2 = y - 17 = z + 3- 3 and x + 2- 1 = 2 y - 84 = z - 54And check whether the lines are parallel or perpendicular.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the angle between the following pair of lines:
$\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4}$
And check whether the lines are parallel or perpendicular.

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.