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Mathematics Question on Plane

The direction cosines of a line passing through two points P(x1,y1,z1)P(x_1, y_1, z_1) and Q(x2,y2,z2)Q(x_2, y_2, z_2) are

A

(x2−x1)\left(x_{2} - x_{1}\right), (y2−y1)\left(y_{2} - y_{1}\right), (z2−z1)\left(z_{2} - z_{1}\right)

B

(x2+x1)\left(x_{2} + x_{1}\right), (y2+y1)\left(y_{2} + y_{1}\right), (z2+z1)\left(z_{2} + z_{1}\right)

C

x2−x1PQ\frac{x_{2}-x_{1}}{PQ}, y2−y1PQ\frac{y_{2}-y_{1}}{PQ}, z2−z1PQ\frac{z_{2}-z_{1}}{PQ}

D

x2+x1PQ\frac{x_{2}+x_{1}}{PQ}, y2+y1PQ\frac{y_{2}+y_{1}}{PQ}, z2+z1PQ\frac{z_{2}+z_{1}}{PQ}

Answer

x2−x1PQ\frac{x_{2}-x_{1}}{PQ}, y2−y1PQ\frac{y_{2}-y_{1}}{PQ}, z2−z1PQ\frac{z_{2}-z_{1}}{PQ}

Explanation

Solution

P(x1,y1,z1)P\left(x_{1}, y_{1}, z_{1}\right) and Q(x2,y2,z2)Q\left(x_{2}, y_{2}, z_{2}\right) ∴\therefore Direction ratios of line PQ=(x2−x1,y2−y1,z2−z1)PQ=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right) ⇒\Rightarrow direction cosine of PQ=PQ = [x2−x1(x2−x1)2+(y2−y1)2+(z2−z1)2\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}}}, y2−y1(x2−x1)2+(y2−y1)2+(z2−z1)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}}}, z2−z1(x2−x1)2+(y2−y1)2+(z2−z1)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] =[x2−x1PQ,y2−y1PQ,z2−z1PQ]=\left[\frac{x_{2}-x_{1}}{PQ}, \frac{y_{2}-y_{1}}{PQ}, \frac{z_{2}-z_{1}}{PQ}\right] where PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}