If theta is the angle between the lines whose | vedclass

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

Question

(A) Given the equations: $6mn - 2nl + 5lm = 0$  $(1)$ $3l + m + 5n = 0 \implies m = -(3l + 5n)$  $(2)$ Substitute $(2)$ into $(1)$: $6n(-(3l + 5n)) -

Vedclass · Accepted Answer

$\frac{\sqrt{35}}{6}$

Vedclass · Answer

(A) Given the equations: $6mn - 2nl + 5lm = 0$  $(1)$ $3l + m + 5n = 0 \implies m = -(3l + 5n)$  $(2)$ Substitute $(2)$ into $(1)$: $6n(-(3l + 5n)) - 2nl + 5l(-(3l + 5n)) = 0$ $-18ln - 30n^2 - 2nl - 15l^2 - 25ln = 0$ $-15l^2 - 45ln - 30n^2 = 0$ Divide by $-15$: $l^2 + 3ln + 2n^2 = 0$ $(l + n)(l + 2n) = 0$ Case $1$: $l = -n$. Then $m = -(3(-n) + 5n) = -2n$. Direction ratios are $(-n, -2n, n)$,i.e.,$(1, 2, -1)$. Case $2$: $l = -2n$. Then $m = -(3(-2n) + 5n) = n$. Direction ratios are $(-2n, n, n)$,i.e.,$(-2, 1, 1)$. Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -2\hat{i} + \hat{j} + \hat{k}$. $\cos 	heta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|} = \frac{|-2 + 2 - 1|}{\sqrt{1+4+1} \sqrt{4+1+1}} = \frac{1}{\sqrt{6} \sqrt{6}} = \frac{1}{6}$. Since $\sin^2 	heta = 1 - \cos^2 	heta = 1 - (\frac{1}{6})^2 = 1 - \frac{1}{36} = \frac{35}{36}$. Therefore,$\sin 	heta = \frac{\sqrt{35}}{6}$.

If $\theta$ is the angle between the lines whose direction cosines $(l, m, n)$ satisfy the equations $6mn - 2nl + 5lm = 0$ and $3l + m + 5n = 0$,then $\sin \theta = $

Similar Questions

If a line makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $x$ and $y$-axes respectively,then the angle made by the line with the positive $z$-axis is

If the projections of a line on the coordinate axes are $2, -1, 2$,then the length of the line is

The vector $\vec{r}$ whose magnitude is $3\sqrt{2}$ units and which makes angles of $\frac{\pi}{4}$ and $\frac{\pi}{2}$ with the positive $y$- and $z$-axes respectively is....

If two lines are parallel to each other,then which of the following is true? (If $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are direction cosines of the two lines).

If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then the value of ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma + {\sin ^2}\delta$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module