If l_1, m_1, n_1 and l_2, m_2, n_2 are the di | vedclass

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

Question

(A) Let the direction vectors of the two perpendicular lines be $\vec{v_1} = (l_1, m_1, n_1)$ and $\vec{v_2} = (l_2, m_2, n_2)$.$A$ line perpendicular

Vedclass · Accepted Answer

$(m_1n_2 - m_2n_1), (n_1l_2 - n_2l_1), (l_1m_2 - l_2m_1)$

Vedclass · Answer

(A) Let the direction vectors of the two perpendicular lines be $\vec{v_1} = (l_1, m_1, n_1)$ and $\vec{v_2} = (l_2, m_2, n_2)$.$A$ line perpendicular to both lines will have a direction vector parallel to the cross product of the direction vectors of the two lines.The cross product $\vec{v_1} 	imes \vec{v_2}$ is given by the determinant:$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \end{vmatrix} = \hat{i}(m_1n_2 - m_2n_1) - \hat{j}(l_1n_2 - l_2n_1) + \hat{k}(l_1m_2 - l_2m_1)$.This simplifies to $\hat{i}(m_1n_2 - m_2n_1) + \hat{j}(n_1l_2 - n_2l_1) + \hat{k}(l_1m_2 - l_2m_1)$.Thus,the direction ratios of the line perpendicular to both are $(m_1n_2 - m_2n_1), (n_1l_2 - n_2l_1), (l_1m_2 - l_2m_1)$.Since the original lines are perpendicular and their direction vectors are unit vectors,the cross product results in a vector whose components represent the direction cosines of the common perpendicular line.

If $l_1, m_1, n_1$ and $l_2, m_2, n_2$ are the direction cosines of two perpendicular lines,then the direction cosines of the line which is perpendicular to both the lines will be:

Similar Questions

Let $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}, \vec{c}=\hat{i}-\hat{j}$ and $\vec{d}=\hat{i}+\hat{j}+x \hat{k}$. If $(\vec{a} \times \vec{b}) \times \vec{c}$ is perpendicular to $\vec{d}$,then $x=$

If $A, B, C, D$ are any four points in space, then $|\overrightarrow{AB} \times \overrightarrow{CD} + \overrightarrow{BC} \times \overrightarrow{AD} + \overrightarrow{CA} \times \overrightarrow{BD}|$ is equal to (where $\Delta$ denotes the area of $\Delta ABC$) (in $\Delta$)

Let $\hat{\alpha}, \hat{\beta}, \hat{\gamma}$ be three unit vectors such that $\hat{\alpha} \times (\hat{\beta} \times \hat{\gamma}) = \frac{1}{2}(\hat{\beta} + \hat{\gamma})$. If $\hat{\beta}$ is not parallel to $\hat{\gamma}$,then the angle between $\hat{\alpha}$ and $\hat{\beta}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module