If the direction cosines of a line are left(f | vedclass

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

Question

(B) The sum of the squares of the direction cosines of a line is always $1$. Given direction cosines are $\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \f

Vedclass · Accepted Answer

$21$

Vedclass · Answer

(B) The sum of the squares of the direction cosines of a line is always $1$. Given direction cosines are $\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}$. Therefore,$\left(\frac{a}{\sqrt{83}}\right)^2 + \left(\frac{5}{\sqrt{83}}\right)^2 + \left(\frac{c}{\sqrt{83}}\right)^2 = 1$. $\Rightarrow \frac{a^2}{83} + \frac{25}{83} + \frac{c^2}{83} = 1$. $\Rightarrow a^2 + 25 + c^2 = 83$. $\Rightarrow a^2 + c^2 = 58$ ...$(i)$. Given $c - a = 4$,squaring both sides gives $(c - a)^2 = 16$. $c^2 + a^2 - 2ca = 16$. Substituting $a^2 + c^2 = 58$ from equation $(i)$: $58 - 2ca = 16$. $2ca = 58 - 16 = 42$. $ca = 21$.

If the direction cosines of a line are $\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$ and $c-a=4$,then $ca=$

Similar Questions

The direction cosines of the line passing through $P(2, 3, -1)$ and the origin are

If a line makes an angle of $\frac{\pi}{3}$ with each of the $X$ and $Y$-axes,then the acute angle made by the line with the $Z$-axis 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).

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

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes in three-dimensional space,then find the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module