The direction cosines of three lines passing | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Three lines passing through the origin with direction cosines $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$ are coplanar if they lie i

Vedclass · Accepted Answer

$\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$

Vedclass · Answer

(A) Three lines passing through the origin with direction cosines $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$ are coplanar if they lie in the same plane. Since they all pass through the origin $(0, 0, 0)$,they are coplanar if and only if the determinant of their direction cosines is zero.This is because the lines are coplanar if there exists a non-zero normal vector $(l, m, n)$ such that it is perpendicular to all three lines.Thus,we have the system of equations:$l l_1 + m m_1 + n n_1 = 0$$l l_2 + m m_2 + n n_2 = 0$$l l_3 + m m_3 + n n_3 = 0$For a non-trivial solution $(l, m, n) 
eq (0, 0, 0)$,the determinant of the coefficient matrix must be zero:$\left| \begin{array}{ccc} l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \ l_3 & m_3 & n_3 \end{array} ight| = 0$By swapping the second and third columns,the sign of the determinant changes,but it remains equal to zero:$-\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$Therefore,$\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$ is the correct condition.

The direction cosines of three lines passing through the origin are $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$. The lines will be coplanar if

Similar Questions

Which of the following statements is false?

$A$ vector $\overline{OP}$ makes angles of $45^{\circ}$ with $OX$ and $60^{\circ}$ with $OY$. Find the angle that $\overline{OP}$ makes with $OZ$ (in degrees).

$E(1,0,0), F(0,2,0), G(0,0,3)$ are respectively the mid-points of the sides $AB, BC, CA$ of $\triangle ABC$. If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are respectively the direction ratios of $AF$ and $BG$,then $\frac{a_1^2+b_1^2+c_1^2}{a_2^2+b_2^2+c_2^2}=$

Find the projection of the line segment joining the points $(-1, 0, 3)$ and $(2, 5, 1)$ on a line whose direction ratios are $6, 2, 3$.

Fill in the blanks:
Coordinate planes divide the space into ........ octants.

Vedclass Test Series

Exam Paper Generator

Online Exam Module