Question:
The direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are
The direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are
Updated On: Jul 7, 2022
- $\left(x_{2} - x_{1}\right)$, $\left(y_{2} - y_{1}\right)$, $\left(z_{2} - z_{1}\right)$
- $\left(x_{2} + x_{1}\right)$, $\left(y_{2} + y_{1}\right)$, $\left(z_{2} + z_{1}\right)$
- $\frac{x_{2}-x_{1}}{PQ}$, $\frac{y_{2}-y_{1}}{PQ}$, $\frac{z_{2}-z_{1}}{PQ}$
- $\frac{x_{2}+x_{1}}{PQ}$, $\frac{y_{2}+y_{1}}{PQ}$, $\frac{z_{2}+z_{1}}{PQ}$
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The Correct Option is C
Solution and Explanation
$P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$
$\therefore$ Direction ratios of line $PQ=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right)$
$\Rightarrow$ direction cosine of $PQ =$
$\bigg[\frac{x_{2}-x_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}$,
$\frac{y_{2}-y_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}$,
$\frac{z_{2}-z_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}\bigg]$
$=\left[\frac{x_{2}-x_{1}}{PQ}, \frac{y_{2}-y_{1}}{PQ}, \frac{z_{2}-z_{1}}{PQ}\right]$
where $PQ=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$
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Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.