3d Coordinate Geometry - Notes, Concept and All Important Formula

3D-COORDINATE GEOMETRY

1. DISTANCE FORMULA:

The distance between two points $A \left( x _{1}, y _{1}, z _{1}\right)$ and $B \left( x _{2}, y _{2}, z _{2}\right)$ is given by $A B=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}\right]}$

All Chapter Notes, Concept and Important Formula

2. SECTION FORMULAE :

Let $P \left( x _{1}, y _{1}, z _{1}\right)$ and $Q \left( x _{2}, y _{2}, z _{2}\right)$ be two points and let $R ( x , y , z )$ divide $PQ$ in the ratio $m _{1}: m _{2}$ . Then $R$ is

$(x, y, z)=\left(\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}, \frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}, \frac{m_{1} z_{2}+m_{2} z_{1}}{m_{1}+m_{2}}\right)$

If $\left( m _{1} / m _{2}\right)$ is positive, $R$ divides $PQ$ internally and if $\left( m _{1} / m _{2}\right)$ is negative, then externally.

Mid point of $PQ$ is given by $\left(\frac{ x _{1}+ x _{2}}{2}, \frac{ y _{1}+ y _{2}}{2}, \frac{ z _{1}+ z _{2}}{2}\right)$

3. CENTROID OF A TRIANGLE :

Let $A \left( x _{1}, y_{1}, z _{1}\right), B \left( x _{2}, y _{2}, z _{2}\right), C \left( x _{3}, y _{3}, z _{3}\right)$ be the vertices of a triangle $ABC$ . Then its centroid $G$ is given by

$G=\left(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}, \dfrac{z_{1}+z_{2}+z_{3}}{3}\right)$

4. DIRECTION COSINES OF LINE :

If $\alpha, \beta, \gamma$ be the angles made by a line with $x$ -axis, $y$ -axis & $z$ -axis respectively then $\cos \alpha, \cos \beta$ & $cos\gamma$ are called direction cosines of a line, denoted by $1, m$ & $n$ respectively and the relation between $\ell$ , $m , n$ is given by $\ell^{2}+ m ^{2}+ n ^{2}=1$

Direction cosine of $x$ -axis, $y$ -axis $\& z$ -axis are respectively $(1,0,0 ; \, 0,1,0 ; \,0,0,1)$

5. DIRECTION RATIOS :

Any three numbers a, b, c proportional to direction cosines $\ell, m , n$ are called direction ratios of the line.

i.e. $\dfrac{\ell}{ a }=\dfrac{ m }{ b }=\dfrac{ n }{ c }$

It is easy to see that there can be infinitely many sets of direction ratios for a given line.

6. RELATION BETWEEN D.C'S & D.R'S:

$\dfrac{\ell}{ a }=\dfrac{ m }{ b }=\dfrac{ n }{ c }$

$\therefore \frac{\ell^{2}}{a^{2}}=\frac{m^{2}}{b^{2}}=\frac{n^{2}}{c^{2}}=\frac{\ell^{2}+m^{2}+n^{2}}{a^{2}+b^{2}+c^{2}}$

$\therefore \ell=\frac{\pm a}{\sqrt{a^{2}+b^{2}+c^{2}}} ; m=\frac{\pm b}{\sqrt{a^{2}+b^{2}+c^{2}}} ; n=\frac{\pm c}{\sqrt{a^{2}+b^{2}+c^{2}}}$

7. DIRECTION COSINE OF AXES :

Direction ratios and Direction cosines of the line joining two points:

Let $A \left( x _{1}, y _{1}, z _{1}\right)$ and $B \left( x _{2}, y _{2}, z _{2}\right)$ be two points, then d.r.'s of $AB$ are $x _{2}- x _{1}, y_{2}- y _{1}, z _{2}- z _{1}$ and the d.c.'s of $AB$ are $\dfrac{1}{ r }\left( x _{2}- x _{1}\right), \dfrac{1}{ r }\left( y _{2}- y _{1}\right)$ $\dfrac{1}{r}\left(z_{2}-z_{1}\right)$ where $r=\sqrt{\left[\Sigma\left(x_{2}-x_{1}\right)^{2}\right]}=|\overrightarrow{A B}|$

8. PROJECTION OF A LINE ON ANOTHER LINE :

Let $PQ$ be a line segment with $P \left( x _{1}, y _{1}, z _{1}\right)$ and $Q \left( x _{2}, y _{2}, z _{2}\right)$ and let $L$ be a straight line whose d.c.'s are $\ell, m , n$ . Then the length of projection of $P Q$ on the $\operatorname{line} L$ is $\left|\ell\left(x_{2}-x_{1}\right)+m\left(y_{2}-y_{1}\right)+n\left(z_{2}-z_{1}\right)\right|$

9. ANGLE BETWEEN TWO LINES :

Let $\theta$ be the angle between the lines with d.c.'s $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}$ , $n _{2}$ then $\cos \theta= l _{1} l _{2}+ m _{1} m _{2}+ n _{1} n _{2} .$ If $a _{1}, b _{1}, c _{1}$ and $a _{2}, b _{2}, c _{2}$ be D.R.'s of two lines then angle $\theta$ between them is given by

$\cos \theta=\dfrac{\left( a _{1} a _{2}+ b _{1} b _{2}+ c _{1} c _{2}\right)}{\sqrt{\left( a _{1}^{2}+ b _{1}^{2}+ c _{1}^{2}\right)} \sqrt{\left( a _{2}^{2}+ b _{2}^{2}+ c _{2}^{2}\right)}}$

10. PERPENDICULARITY AND PARALLELISM:

Let the two lines have their d.c.'s given by $l _{1}, m _{1}, n _{1}$ and $l _{2}, m _{2}, n _{2}$ respectively then they are perpendicular if $\theta=90^{\circ}$ i.e. $\cos \theta=0$ , i.e. $l _{1} l _{2}+ m _{1} m _{2}+ n _{1} n _{2}=0$

Also the two lines are parallel if $\theta=0$ i.e. $\sin \theta=0$ , i.e. $\dfrac{\ell_{1}}{\ell_{2}}=\dfrac{ m _{1}}{ m _{2}}=\dfrac{ n _{1}}{ n _{2}}$

Note:

If instead of d.c.'s, d.r.'s $a _{1}, b _{1}, c _{1}$ and $a _{2}, b _{2}, c _{2}$ are given, then the lines are perpendicular if $a _{1} a _{2}+ b _{1} b _{2}+ c _{1} c _{2}=0$ and parallel if $a _{1} / a _{2}= b _{1} / b _{2}= c _{1} / c _{2}$

11. EQUATION OF A STRAIGHT LINE IN SYMMETRICAL FORM :

(a) One point form : Let $A\left(x_{1}, y_{1}, z_{1}\right)$ be a given point on the straight line and $l , m , n$ the $d$ .c's of the line, then its equation is

$\dfrac{ x - x _{1}}{\ell}=\dfrac{ y - y _{1}}{ m }=\dfrac{ z - z _{1}}{ n }= r \quad \text { (say) }$

It should be noted that $P \left( x _{1}+ lr , y _{1}+ mr , z _{1}+ nr \right)$ is a general point on this line at a distance $r$ from the point $A \left( x _{1}, y _{1}, z _{1}\right)$ i.e. $AP = r$ . One should note that for $AP = r ; 1, m , n$ must be d.c.'s not d.r.'s. If $a , b , c$ are direction ratios of the line, then equation of the line is $\dfrac{x-x_{1}}{a}=\dfrac{y-y_{1}}{b}=\dfrac{z-z_{1}}{c}=r$ but here $A P \neq r$ .

(b) Equation of the line through two points $A(x_1,y_1, z_1)$ and $B(x_2, y_2, z_2)$ is $\dfrac{x-x_{1}}{x_{2}-x_{1}}=\dfrac{y-y_{1}}{y_{2}-y_{1}}=\dfrac{z-z_{1}}{z_{2}-z_{1}}$

12. FOOT, LENGTH AND EQUATION OF PERPENDICULAR FROM A POINT TO A LINE:

Let equation of the line be

$\dfrac{ x - x _{1}}{\ell}=\dfrac{ y - y _{1}}{ m }=\dfrac{ z - z _{1}}{ n }= r$ . (say) ......(i)

and $A (\alpha, \beta, \gamma)$ be the point. Any point on the line (i) is

$P \left(\ell r + x _{1}, mr + y _{1}, nr + z _{1}\right)$ . .........(ii)

If it is the foot of the perpendicular, from $A$ on the line, then $A P$ is $\perp$ to the line, so $\scriptsize{\ell\left(\ell r + x _{1}-\alpha\right)+ m \left( mr + y _{1}-\beta\right)+ n \left( nr + z _{1}-\gamma\right)=0}$

i.e. $\quad r=\left(\alpha-x_{1}\right) \ell+\left(\beta-y_{1}\right) m+\left(\gamma-z_{1}\right) n$

since $\quad \ell^{2}+m^{2}+n^{2}=1$

Putting this value of $r$ in (ii), we get the foot of perpendicular from point $A$ to the line.

Length : Since foot of perpendicular $P$ is known, length of perpendicular,

$\scriptsize{A P=\sqrt{\left[\left(\ell r+x_{1}-\alpha\right)^{2}+\left(m r+y_{1}-\beta\right)^{2}+\left(n r+z_{1}-\gamma\right)^{2}\right]}}$

Equation of perpendicular is given by

$\dfrac{ x -\alpha}{\ell r + x _{1}-\alpha}=\dfrac{ y -\beta}{ mr + y _{1}-\beta}=\dfrac{ z -\gamma}{ nr + z _{1}-\gamma}$

13. EQUATIONS OF A PLANE :

The equation of every plane is of the first degree i.e. of the form $a x+b y+c z+d=0$ , in which $a, b, c$ are constants, where $a^{2}+b^{2}+c^{2} z$ 0 (i.e. $a , b , c \neq 0$ simultaneously).

(a) Vector form of equation of plane:

If $\vec{a}$ be the position vector of a point on the plane and $\vec{n}$ be a vector normal to the plane then it's vectorial equation is given by $(\overrightarrow{ r }-\overrightarrow{ a }) \cdot \overrightarrow{ n }=0 \Rightarrow \overrightarrow{ r } \cdot \overrightarrow{ n }= d$ where $d =\overrightarrow{ a } \cdot \overrightarrow{ n }= constant$

(b) Plane Parallel to the Coordinate Planes:

(i) Equation of $y$ -z plane is $x=0$ .

(ii) Equation of $z$ -x plane is $y=0$ .

(iii) Equation of $x$ -y plane is $z=0$ .

(iv) Equation of the plane parallel to $x$ -y plane at a distance $c$ is $z=c$ . Similarly, planes parallel to $y$ -z plane and $z$ -x plane are respectively $x=c$ and $y=c$ .

(c) Equations of Planes Parallel to the Axes:

If $a=0$ , the plane is parallel to $x$ -axis i.e. equation of the plane parallel to $x$ -axis is $by + cz + d =0$ .

Similarly, equations of planes parallel to $y$ -axis and parallel to $z$ -axis are $ax + cz + d =0$ and $ax + by + d =0$ respectively.

(d) Equation of a Plane in Intercept Form:

Equation of the plane which cuts off intercepts $a , b , c$ from the axes is $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ .

(e) Equation of a Plane in Normal Form:

If the length of the perpendicular distance of the plane from the origin is $p$ and direction cosines of this perpendicular are $( l$ $m , n$ ), then the equation of the plane is $lx + m y+ nz = p$

(f) Vectorial form of Normal equation of plane :

If $\hat{n}$ is a unit vector normal to the plane from the origin to the plane and d be the perpendicular distance of plane from origin then its vector equation is $\overrightarrow{ r } \cdot \hat{ n }= d$ .

(g) Equation of a Plane through three points:

The equation of the plane through three non-collinear points

$\left( x _{1}, y _{1}, z_{1}\right),\left( x _{2}, y_{2}, z _{2}\right)\left( x _{3}, y_{3}, z _{3}\right)$ is $\left|\begin{array}{llll} x & y & z & 1 \\ x _{1} & y _{1} & z _{1} & 1 \\ x _{2} & y_{2} & z _{2} & 1 \\ x _{3} & y _{3} & z _{3} & 1\end{array}\right|=0$

14. ANGLE BETWEEN TWO PLANES:

Consider two planes $a x+b y+c z+d=0$ and $a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$

Angle between these planes is the angle between their normals.

$\cos \theta=\dfrac{a a^{\prime}+b b^{\prime}+c c^{\prime}}{\sqrt{a^{2}+b^{2}+c^{2}} \sqrt{a^{\prime 2}+b^{\prime 2}+c^{\prime 2}}}$

$\therefore$ Planes are perpendicular if $a a^{\prime}+b b^{\prime}+c c^{\prime}=0$ and they are parallel if a/a' = b/b' = c/c'.

Planes parallel to a given Plane:

Equation of a plane parallel to the plane $a x+b y+c z+d=0$ is $a x+b y+c z+d^{\prime}=0 .$ d' is to be found by other given condition.

15. ANGLE BETWEEN A LINE AND A PLANE :

Let equations of the line and plane be $\dfrac{ x - x _{1}}{\ell}=\dfrac{ y - y _{1}}{ m }=\dfrac{ z - z _{1}}{ n }$ and $a x+b y+c z+d=0$ respectively and $\theta$ be the angle which line makes with the plane. Then $(\pi / 2-\theta)$ is the angle between the line and the normal to the plane.

So $\sin \theta=\dfrac{ a \ell+ bm + cn }{\sqrt{\left( a ^{2}+ b ^{2}+ c ^{2}\right)} \sqrt{\left(\ell^{2}+ m ^{2}+ n ^{2}\right)}}$

Line is parallel to plane if $\theta=0$

i.e. if $a l+b m+c n=0$

Line is $\mathbf \perp$ to the plane if line is parallel to the normal of the plane

i.e. if $\dfrac{ a }{\ell}=\dfrac{ b }{ m }=\dfrac{ c }{ n }$ .

16. CONDITION IN ORDER THAT THE LINE MAY LIE ON THE GIVEN PLANE :

The line $\dfrac{ x - x _{1}}{\ell}=\dfrac{ y - y _{1}}{ m }=\dfrac{ z - z _{1}}{ n }$ will ie on the plane $Ax + By + Cz + D =0$ if (a) $A \ell+B m+C n=0$ and $( b ) A x_{1}+B y_{1}+C z_{1}+D=0$

17. POSITION OF TWO POINTS W.R.T. A PLANE :

Two points $P \left( x _{1}, y _{1}, z _{1}\right) \& Q \left( x _{2}, y _{2}, z _{2}\right)$ are on the same or opposite sides of a plane $a x+b y+c z+d=0$ according to $a x_{1}+b y_{1}+c z_{1}+d$ & $ax _{2}+ by _{2}+ cz _{2}+ d$ are of same or opposite signs.

18. IMAGE OF A POINT IN THE PLANE :

Let the image of a point $P \left( x _{1}, y _{1}, z _{1}\right)$ in a plane $a x+b y+c z+d=0$ is $Q \left( x _{2}, y _{2}, z _{2}\right)$ and foot of perpendicular of point $P$ on plane is $R \left( x _{3}, y _{3}, z _{3}\right)$ , then

(a) $\scriptsize{\dfrac{ x _{3}- x _{1}}{ a }=\dfrac{y_{3}- y _{1}}{ b }=\dfrac{ z _{3}- z _{1}}{ c }=-\left(\dfrac{ ax _{1}+ by _{1}+ cz _{1}+ d }{ a ^{2}+ b ^{2}+ c ^{2}}\right)}$

(b) $\scriptsize{\dfrac{ x _{2}- x _{1}}{ a }=\dfrac{ y _{2}- y _{1}}{ b }=\dfrac{ z _{2}- z _{1}}{ c }=-2\left(\dfrac{ ax _{1}+ by _{1}+ cz _{1}+ d }{ a ^{2}+ b ^{2}+ c ^{2}}\right)}$

19. CONDITION FOR COPLANARITY OF TWO LINES :

Let the two lines be

$\dfrac{ x -\alpha_{1}}{\ell_{1}}=\dfrac{y-\beta_{1}}{ m _{1}}=\dfrac{z-\gamma_{1}}{ n _{1}}$ .......(i)

and $\dfrac{ x -\alpha_{2}}{\ell_{2}}=\dfrac{y-\beta_{2}}{ m _{2}}=\dfrac{z-\gamma_{2}}{ n _{2}}$ ........(ii)

These lines will coplanar if $\left|\begin{array}{ccc}\alpha_{2}-\alpha_{1} & \beta_{2}-\beta_{1} & \gamma_{2}-\gamma_{1} \\ \ell_{1} & m _{1} & n _{1} \\ \ell_{2} & m _{2} & n _{2}\end{array}\right|=0$

the plane containing the two lines is $\left|\begin{array}{ccc} x -\alpha_{1} & y -\beta_{1} & z -\gamma_{1} \\ \ell_{1} & m _{1} & n _{1} \\ \ell_{2} & m _{2} & n _{2}\end{array}\right|=0$

20. PERPENDICULAR DISTANCE OF A POINT FROM THE PLANE :

Perpendicular distance $p$ , of the point $A \left( x _{1}, y _{1}, z _{1}\right)$ from the plane $a x+b y+c z+d=0$ is given by

$p =\dfrac{\left| ax _{1}+ by _{1}+ cz _{1}+ d \right|}{\sqrt{\left( a ^{2}+ b ^{2}+ c ^{2}\right)}}$

Distance between two parallel planes $a x+b y+c z+d_{1}=0$ & $a x+b y+c z+d_{2}=0$ is

$\left|\dfrac{d_{1}-d_{2}}{\sqrt{a^{2}+b^{2}+c^{2}}}\right|$

21. A PLANE THROUGH THE LINE OF INTERSECTION OF TWO GIVENPLANES:

Consider two planes

$u \equiv a x+b y+c z+d=0$ and $v \equiv a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$

The equation $u +\lambda v =0, \lambda$ a real parameter, represents the plane passing through the line of intersection of given planes and if planes are parallel, this represents a plane parallel to them.

22. BISECTORS OF ANGLES BETWEEN TWO PLANES :

Let the equations of the two planes be $a x+b y+c z+d=0$ and $a_{1} x+b_{1} y+c_{1} z+d_{1}=0$

Then equations of bisectors of angles between them are given by

$\dfrac{a x+b y+c z+d}{\sqrt{\left(a^{2}+b^{2}+c^{2}\right)}}=\pm \dfrac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{\left(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\right)}}$

(a) Equation of bisector of the angle containing origin : First make both constant terms positive. Then +ve sign give the bisector of the angle which contains the origin.

(b) Bisector of acute/obtuse angle : First making both constant terms positive,

$aa _{1}+ bb _{1}+ cc _{1}>0 \quad \Rightarrow$ origin lies in obtuse angle

$aa _{1}+ bb _{1}+ cc _{1}<0 \quad \Rightarrow$ origin lies in acute angle

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