1 Marks Question

Question 11 Mark

Write the condition of three lines with direction cosines $l_1, m_1, n_1 ; l_2, m_2, n_2$ and $l_3, m_3, n_3$ as coplanar.

Answer

The required condition is :
$
\left|\begin{array}{lll}
l_1 & m_1 & n_1 \\
l_2 & m_2 & n_2 \\
l_3 & m_3 & n_3
\end{array}\right|=0
$

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Question 21 Mark

If the direction ratios of line $A B$ are $\cos \alpha, \cos \beta, \cos \gamma$, then what will be the direction cosines of line BA?

Answer

Direction cosines will be
$\cos (\pi-\alpha), \cos (\pi-\beta), \cos (\pi-\gamma)$
i.e., $-\cos \alpha,-\cos \beta,-\cos \gamma$

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Question 31 Mark

The direction ratios of two mutually perpendicular lines are $1,2,3$ and $3,2, \lambda$. Then write the value of $\lambda$.

Answer

For being perpendicular
$
\begin{array}{rlrl}
a_1 a_2+b_1 b_2+c_1 c_2 =0 \\
1 \times 3+2 \times 2+3 \times \lambda =0 \\
\Rightarrow 3+4+3 \lambda =0 \\
\therefore 3 \lambda =-7 \\
\Rightarrow \lambda =-\frac{7}{3} \quad \text { }
\end{array}
$

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Question 41 Mark

Find the direction cosines of the line joining the points $(1,0,0)$ and $(0,1,1)$.

Answer

Directions ratios of the line will be
$0-1,1-0,1-0$
$\text { i.e., }-1,1,1$
Hence direction cosines of the line are
$\frac{-1}{\sqrt{(-1)^2+(1)^2+(1)^2}}, \frac{1}{\sqrt{(-1)^2+(1)^2+(1)^2}}, \frac{1}{\sqrt{(-1)^2+(1)^2+(1)^2}}$
i.e., $\frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

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