Three Dimensional Geometry - Revision Notes

 CBSE Class 12 Mathematics

Chapter-11
Three Dimensional Geometry

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• Direction cosines of a line : Direction cosines of a line are the cosines of the angles made by the line with the positive direct ions of the coordinate axes.
• are the direct ion cosines of a line, then  
• Direct ion cosines of a line joining two pointsandare 

·   where  

• Direction ratios of a line are the numbers which are proportional to the direct ion cosines of a line.
• If  are the direct ion cosines and   are the direct ion ratios of a line

Then, ,,

• Skew lines: Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.
• Angle between two skew lines: Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.
• If are the direction cosines of two lines; and  is the acute angle between the two lines; then,

                   cosθ=∣∣∣a1a2+b1b2+c12a21+b21+c21√a22+b22+c22√∣∣∣$\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+{c_1}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|$

• Vector equation of a line that passes through the given point whose position vector is   and parallel to a given vector  
• Equation of a line through a point    and having direct ion cosines   is 
• The vector equation of a line which passes through two points whose position vectors are  
• Cartesian equation of a line that passes through two points  and  is 
• If  is the acute angle between  and    then, cosθ=∣∣∣∣b1→b2→∣∣∣b1→∣∣∣∣∣∣b2→∣∣∣∣∣∣∣$\cos \theta =\left|\frac{\overrightarrow{b_1}\overrightarrow{b_2}}{\left|\overrightarrow{b_1}\right|\left|\overrightarrow{b_2}\right|}\right|$
• If    and  are the equations of two lines, then the acute angle between the two lines is given by 
• Shortest distance between two skew lines is the line segment perpendicular to both the lines.
• Shortest distance between r→=a1→+λb1→$\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and r→=a2→+λb2→$\overrightarrow{r}=\overrightarrow{a_2}+\lambda \overrightarrow{b_2}$ is ∣∣∣∣(b1→×b2→).(a2→−a1→)|b1×b2|∣∣∣∣$\left|\frac{\left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right).\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)}{\left|b_1\times b_2\right|}\right|$
• Shortest distance between the lines:    and    is

• Distance between parallel lines  r→=a1→+λb1→$\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and r→=a2→+λb2→$\overrightarrow{r}=\overrightarrow{a_2}+\lambda \overrightarrow{b_2}$is ∣∣∣∣b→×(a2→−a1→)|b|∣∣∣∣$\left|\frac{\overrightarrow{b}\times \left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)}{\left|b\right|}\right|$
• In the vector form, equation of a plane which is at a distance d from the origin, and nˆ$\hat{n}$is the unit vector normal to the plane through the origin is 
• Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n is .
• The equation of a plane through a point whose position vector is a and perpendicular to the vector N→$\overrightarrow{\mathrm{N}}$ is (r→−a→).N→=0$\left(\overrightarrow{r}-\overrightarrow{a}\right).\overrightarrow{\mathrm{N}}=0$.
• Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point is 
• Equation of a plane passing through three non collinear points ,   
• Vector equation of a plane that contains three non collinear points having position vectors a→,b→$\overrightarrow{a},\overrightarrow{b}$and c→$\overrightarrow{c}$ is  (r→−a→).[(b→−a→)×(c→−a→)]=0$\left(\overrightarrow{r}-\overrightarrow{a}\right).\left[\left(\overrightarrow{b}-\overrightarrow{a}\right)\times \left(\overrightarrow{c}-\overrightarrow{a}\right)\right]=0$.
• Equation of a plane that cuts the coordinates axes at  is .
• Vector equation of a plane that passes through the intersection of planes r→.n1−→=d1$\overrightarrow{r}.\overrightarrow{n_1}=d_1$ and r→.n2−→=d2$\overrightarrow{r}.\overrightarrow{n_2}=d_2$ is r→(n1−→−λn2−→)=d1+λd2$\overrightarrow{r}\left(\overrightarrow{n_1}-\lambda \overrightarrow{n_2}\right)=d_1+\lambda d_2$,  where  is any non-zero constant.
• Cartesian equation of a plane that passes that passes through the intersection of two given planes  and is
• Two lines r→=a1→+λb1→$\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and r→=a2→+μb2→$\overrightarrow{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2}$ are coplanar if (a2→−a1→).(b1→×b2→)=0.$\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right).\left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right)=0.$
• Two planes a1x+b1y+c1z+d1=0$a_{1}x+b_{1}y+c_{1}z+d_1=0$ and a2x+b2y+c2z+d2=0$a_{2}x+b_{2}y+c_{2}z+d_2=0$   are coplanar if  
• In the vector form, if  is the angle between the two planes, r→.n1−→=d1$\overrightarrow{r}.\overrightarrow{n_1}=d_1$ and r→.n2−→=d2$\overrightarrow{r}.\overrightarrow{n_2}=d_2$, then θ=cos−1∣∣∣n1−→.n2−→n1−→∣∣n2−→∣∣∣∣∣$\theta ={\cos }^{-1}\left|\frac{\overrightarrow{n_1}.\overrightarrow{n_2}}{\overrightarrow{n_1}\left|\overrightarrow{n_2}\right|}\right|$
• The angle  between the line r→=a→+λb→$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and the plane r→.nˆ=d$\overrightarrow{r}.\hat{n}=d$ is sinϕ=b→.nˆ∣∣∣b→∣∣∣∣∣nˆ∣∣$\sin \varphi =\frac{\overrightarrow{b}.\hat{n}}{\left|\overrightarrow{b}\right|\left|\hat{n}\right|}$
• The angle  between the planes  and  is given by 
• The distance of a point whose position vector is a→$\overrightarrow{a}$ from the plane r→.nˆ=d$\overrightarrow{r}.\hat{n}=d$  is ∣∣d−a→.nˆ∣∣$\left|d-\overrightarrow{a}.\hat{n}\right|$.
• The distance from a point  to the plane Ax + By + Cz + D = 0 is 
• Equation of any plane that is parallel to a plane that is parallel to a plane Ax + By + Cz + D = 0 is Ax + By + Cz + k = 0, where k is a different constant other than D.