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 \(A B=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}\right]}\)
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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