We know that if position vector of two points A and B are given as \(\overrightarrow {OA} \) and \(\overrightarrow {OB} \) then
\[ \Rightarrow |\overrightarrow {AB} | = |\overrightarrow {AB} - \overrightarrow {OA} |\] \[ \Rightarrow |\overrightarrow {AB} | = |({x_2}\hat i + {y_2}\hat j + {z_2}\hat k) - ({x_1}\hat i + {y_1}\hat j + {z_1}\hat k)|\] \[ \Rightarrow |\overrightarrow {AB} | = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} \]
7. DISTANCE OF A POINT P FROM COORDINATE AXESLet PA, PB and PC are distances of the point P(x, y, z) from the coordinate axes OX, OY and OZ respectively, then \(PA = \sqrt {{y^2} + {z^2}} ,\;\;PB = \sqrt {{z^2} + {x^2}} ,\;\;PC = \sqrt {{x^2} + {z^2}} \)
8. SECTION FORMULA(i) Internal Division : If point P divides the distance between the points A (x₁, y₁, z₁,) and B (x₂, y₂, Z₂,) in the ratio of m: n (internally). The coordinate of P is given as
Note:All these formulae are very much similar to two dimension coordinate geometry.
9. CENTROID OF A TRIANGLE
10. INCENTRE OF TRIANGLE ABC
11. CENTROID OF A TETRAHEDRON
12. RELATION BETWEEN TWO LINES Two lines in the space may be coplanar and may be none coplanar. Non coplanar lines are called skew lines if they never intersect each other. Two parallel lines are also non intersecting lines but they are coplanar. Two lines whether intersecting or non intersecting, the angle between them can be obtained.
13. DIRECTION COSINES AND DIRECTION RATIOS Direction cosines : Let
a, ß,
y be the angles which directed line makes with the positive directions of the axes of x, y and z respectively, the \(\cos \alpha ,\cos \beta ,\cos \gamma \) are called the direction cosines of the line. The direction cosine denoted
(l, m, n).
(ii) If
l, m, n, be the direction cosines of a lines, then
l²+ m² + n² = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines,
l, m, n, then a, b, c are called the direction ratios.
If a, b, c are the direction ratio of any line L the \(a\hat i + b\hat j + c\hat k\) will be a vector parallel to the line L.
If l, m, n are direction cosine of line L then \(l\hat i + m\hat j + n\hat k\) is a unit vector parallel to the line L.
(iv) If
l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then
(v) If OP = r, when O is the origin and the direction cosines of OP are
l, m, n then the coordinates of P are (
lr, mr, nr).
If direction cosine of the line AB are l, m, n, | AB |= r, and the coordinate of A is (x₁, y₁, z₁,) then the coordinate of B is given as (x₁, + rl, y₁, + rm, z₁ + rn)
(vi) If the coordinates P and Q are (x₁, y₁, z₁) and (x₂, y₂, z₂) then the direction ratios of line PQ are, a = x₂ − x₁, b = y₂ - y₁ and c = z₂ - z₁, and the direction cosines of line PQ are
l = \(\dfrac{{{x_2} - {x_1}}}{{|\overrightarrow {PQ} |}}\), m = \(\dfrac{{{y_2} - {y_1}}}{{|\overrightarrow {PQ} |}}\) and n = \(\dfrac{{{z_2} - {z_1}}}{{|\overrightarrow {PQ} |}}\)
(vii) Direction cosines of axes : Since the positive x-axis makes angles 0°, 90°, 90° with axes of x, y and z respectively. Therefore
Direction cosines of x-axis are (1,0, 0)
Direction cosines of y-axis are (0, 1, 0)
Direction cosines of z-axis are (0, 0, 1)
14. ANGLE BETWEEN TWO LINE SEGMENTS
If two lines having direction ratios a₁, b₁, c₁, and a₂, b₂, c₂, respectively then we can consider two vector parallel to the lines as \({a_1}\hat i + {b_1}\hat j + {c_1}\hat k\) and \({a_2}\hat i + {b_2}\hat j + {c_2}\hat k\) and angle between them can be given as: \[\cos \theta = \dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \cdot \sqrt {a_2^2 + b_2^2 + c_2^2} }}\]
(i) The line will be perpendicular if a1a2 + b1b2 + c1c2 = 0
(ii) The lines will be parallel, if \(\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\).
(iii) Two parallel lines have same direction cosines i.e. l1 = l2, m1 = m2 and n1 = n2.
15. PROJECTION OF A LINE SEGMENT ON A LINE (i) If the coordinates P and Q are (x₁, Y₁, Z₁,) and (X₂, Y₂, Z₂) then the projection of the line segments PQ on a line having direction cosines l, m, n is |l(x₂ - x₁) + m(y₂ - y₁) + n(z₂ - z₁)|
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this surface is called a plane. This fixed line is called the normal to the plane.
16. EQUATION OF A PLANE (i) Normal form of the equation of a plane is lx + my + nz = p, where, l, m, n are the direction cosines of the normal to the plane and p is the distance of the plane from the origin.
(ii) General form : ax + by + cz + d = 0 is the equation of a plane, where a, b, c are the direction ratios of the normal to the plane.
(iii) The equation of a plane passing through the point (x₁, y₁, z₁) is given by \[a({x_1} - {x_2}) + b({y_1} - {y_2}) + c({z_1} - {z_2}) = 0\] where a, b, c are the direction ratios of the normal to the plane.
(iv) Plane through three points : The equation of the plane through three non-collinear points (x1, Y1, z1), (x2, y2, z2) and (x3, y3, z3) is: \[\left| {\begin{array}{*{20}{c}}x&y&z&1\\{{x_1}}&{{y_1}}&{{z_l}}&1\\{{x_2}}&{{y_2}}&{{z_2}}&1\\{{x_3}}&{{y_3}}&{{z_3}}&1\end{array}} \right| = 0\]
(v) Intercept Form: The equation of a plane cutting intercept a, b, c on the axes is \(\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1\)
(vi) Vector Form : The equation of a plane passing through a point having position vector \({\vec a}\) and normal to vector \({\vec n}\) is \((\vec r - \vec a) \cdot \vec n\) or \(\vec r \cdot \vec n = \vec a \cdot \vec n\).
Note: (a) Vector equation of a plane normal to unit vector \({\hat n}\) and at a distance d from the origin is: \(\vec r \cdot \hat n = d\)
(b) Planes parallel to the coordinate planes (i) Equation of yz - plane is x = 0
(ii) Equation of xz - plane is y = 0
(iii) Equation of xy - plane is z = 0
(c) Planes parallel to the axes : If a = 0, the plane is parallel to x-axis i.e. equation of the plane parallel to the x-axis is by + cz + d = 0. Similarly, equation of planes parallel to y-axis and parallel to z-axis are ax + cz + d = 0 and ax + by + d = 0 respectively.
(d) Plane through origin : Equation of plane passing through origin is ax + by + cz = 0.
(e) Transformation of the equation of a plane to the normal form: To reduce any equation ax + by + cz - d = 0 to the normal form, first write the constant term on the right hand side and make it positive, then divided each term by \(\sqrt {{a^2} + {b^2} + {c^2}} \), where a, b, c are coefficients of x, y and z respectively \[\dfrac{{ax}}{{ \pm \sqrt {{a^2} + {b^2} + {c^2}} }} + \dfrac{{by}}{{ \pm \sqrt {{a^2} + {b^2} + {c^2}} }} + \dfrac{{cz}}{{ \pm \sqrt {{a^2} + {b^2} + {c^2}} }} = \dfrac{d}{{ \pm \sqrt {{a^2} + {b^2} + {c^2}} }}\]
Where (+) sign is to be taken if d > 0 an (-) sign is to be taken if d < 0.
(f) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz + λ = 0 distance between two parallel planes ax + by + cz + d1 = 0 and ax + dy + xz + d2 = 0 is given as: \[\frac{{|{d_1} - {d_2}|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
(g) Equation of a plane passing through a given point and parallel to the given vectors: The equation of a plane passing through a point and having position vector \({\vec a}\) and parallel to \({\vec b}\) and \({\vec c}\) is \[\vec r = \vec a + \lambda \vec b + \mu \vec c\] parametric form (where \(\lambda \) and \(\mu \) are scalers).
OR
\(\vec r \cdot (\vec b \times \vec c) = \vec a \cdot (\vec b \times \vec c)\) (non parametric form)
(h) A plane ax + by + cz + d = 0 divides the line segment joining (x1, y1, z1) and (x2, y2, z2) in the ratio: \(\left( { - \dfrac{{a{x_1} + b{y_1} + c{z_1} + d}}{{a{x_2} + b{y_2} + c{z_2} + d}}} \right)\)
(i) The xy-plane divides the line segment joining the point (x1, y1, z1) and (x2, y2, z2) in the ratio \( - \dfrac{{{z_1}}}{{{z_2}}}\). Similarly yz - plane divides in \( - \dfrac{{{x_1}}}{{{x_2}}}\) and zx - plane divides in \( - \dfrac{{{y_1}}}{{{y_2}}}\).
17. ANGLE BETWEEN TWO PLANES(i) Consider two planes ax + by + cz + d = 0 and a'x + b'y + c'z+ d' = 0. Angle between these planes is the angle between their normal. Since direction ratios of their normal are (a, b, c) and (a', b', c') respectively, hence \(\theta \) the angle between them is given by: \[\cos \theta = \frac{{aa' + bb' + cc'}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {a{'^2} + b{'^2} + c{'^2}} }}\]
Planes are perpendicular if aa' + bb' + c'' =0 and planes are parallel, if \(\dfrac{a}{{a'}} = \dfrac{b}{{b'}} = \dfrac{c}{{c'}}\).
(ii) The angle \(\theta \) between the plane \(\vec r \cdot {{\vec n}_1} = {d_1}\) and \(\vec r \cdot {{\vec n}_1} = {d_1}\) is given by \[\cos \theta = \dfrac{{{{\vec n}_1} \cdot {{\vec n}_2}}}{{|{{\vec n}_1}||{{\vec n}_2}|}}\]
Planes are perpendicular if \({{{\vec n}_1} \cdot {{\vec n}_2} = 0}\) and planes are parallel if \({{{\vec n}_1} = \lambda {{\vec n}_2}}\) .
18. A PLANE AND A POINT (i) Distance of the point (x', y', z') from the plane ax + by + cz + d = 0 is given by \[\dfrac{{ax' + by' + cz' + d}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
(ii) The length of the perpendicular from a point having position vector \({\vec n}\) to the plane \(\vec r \cdot {\vec n} = {d}\) is given by \[p = \dfrac{{|\vec a \cdot \vec n - d|}}{{|\vec n|}}\]
19. ANGLE BISECTORS(i) The equations of the planes bisecting the angle between two given planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are\[\dfrac{{{a_1}x + {b_1}y + {c_1}z + {d_1}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} }} = \pm \dfrac{{{a_2}x + {b_2}y + {c_2}z + {d_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}\]
(ii) Equation of bisector of the angle containing origin: First make both the constant terms positive. then the positive sign in \(\dfrac{{{a_1}x + {b_1}y + {c_1}z + {d_1}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} }} = \pm \dfrac{{{a_2}x + {b_2}y + {c_2}z + {d_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}\) gives the bisector of the angle which contains the origin.
(iii) Bisector of acute/obtuse angle : First make both the constant terms positive. Then
a1a2 + b1b2 + c1c2 > 0 ⇒ origin lies obtuse angle
a1a2 + b1b2 + c1c2 < 0 ⇒ origin lies in acute angle
20. FAMILY OF PLANES(i) Any plane passing through the line of intersection of non- parallel planes OR equation of the plane through the given line in non symmetrical form. a1x + b1y + c1z + d1 =0 and a2x + b2y + c2z + d2 = 0 is a1x + b1y + c1z+ d1 + λ(a2x + b2y + c2z + d2) = 0
(ii) The equation of plane passing through the intersection of the planes \(\vec r \cdot {{\vec n}_1} = {d_1}\) and \(\vec r \cdot {{\vec n}_2} = {d_2}\) is \(\vec r \cdot ({{\vec n}_1} + \lambda {{\vec n}_2}) = {d_1} + \lambda {d_2}\).
(iii) Plane through a given line : Equation of any plane through the line in symmetrical form \(\dfrac{{x - {x_1}}}{l} = \dfrac{{y - {y_1}}}{m} = \dfrac{{z - {z_1}}}{n}\) is A(x - x1) + B(y - y1) + c(z - z1) = 0, where Al + Bm + Cn = 0.
21. AREA OF A TRIANGLELet A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) be the vertices of a triangle then Area of triangle is \(\Delta = \sqrt {\Delta _x^2 + \Delta _y^2 + \Delta _z^2} \), where \({\Delta _x} = \dfrac{1}{2}\left| {\begin{array}{*{20}{c}}{{y_1}}&{{z_1}}&1\\{{y_2}}&{{z_2}}&1\\{{y_3}}&{{z_3}}&1\end{array}} \right|\), \({\Delta _y} = \dfrac{1}{2}\left|
{\begin{array}{*{20}{c}}{{z_1}}&{{x_1}}&1\\{{z_2}}&{{x_2}}&1\\{{z_3}}&{{x_3}}&1\end{array}}
\right|\) and \({\Delta _z} = \dfrac{1}{2}\left|
{\begin{array}{*{20}{c}}{{x_1}}&{{y_1}}&1\\{{x_2}}&{{y_2}}&1\\{{x_3}}&{{y_3}}&1\end{array}}
\right|\)
Vector Method: From Two vectors \(\overrightarrow {AB} \) and \(\overrightarrow {AC} \). then area is given by \[\dfrac{1}{2}|\overrightarrow {AB} \times \overrightarrow {AC} | = \dfrac{1}{2}\left| {\begin{array}{*{20}{c}}{\hat i}&{\hat j}&{\hat k}\\{{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}}\\{{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}}\end{array}} \right|\]
22. VOLUME OF A TETRAHEDRON Volume of a tetrahedron with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, y3) and D(x4, y4, y4,) is given by: \[V = \frac{1}{6}\left| {\begin{array}{*{20}{c}}{{x_1}}&{{y_1}}&{{z_1}}&1\\{{x_2}}&{{y_2}}&{{z_2}}&1\\{{x_3}}&{{y_3}}&{{z_3}}&1\\{{x_4}}&{{y_4}}&{{z_4}}&1\end{array}} \right|\]
23. EQUATION OF A LINE
(i) A straight line in space is characterized by the intersection of two planes which are not parallel and therefore, the equation of a straight line is a solution of the system constituted by the equations of the two planes, a1x + b1y + c1z +d1 = 0 and a2x + b2y + c2z + d2 = 0. This form is also known as non-symmetrical form.
(ii) The equation of a line passing through the point (x1, y1, z1,) and having direction ratios a, b, c is \[\frac{{x - {x_1}}}{a} = \frac{{y - {y_1}}}{b} = \frac{{z - {z_1}}}{c} = r\] This is called the symmetric form. A general point on the line is given by (x1 + ar, y1 + br, z1 + cr).
(iii) Vector Equation: Vector equation of a straight line passing through a fixed point with position vector \({\vec a}\) and parallel to a given vector \({\vec b}\) is \(\vec r = \vec a + \lambda \vec b\) where \(\lambda \) is scalar.
(iv) The equation of the line passing through the points (x1, y1, z1) and (x2, y2, z2) is: \[\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}\]
(v) Vector equation of a straight line passing through two points with position vectors \({\vec a}\) and \({\vec b}\) is \(\vec r = \vec a + \lambda (\vec b - \vec a)\).
(vi) Reduction of cartesion form of equation of a line to vector form and vice versa \[\frac{{x - {x_1}}}{a} = \frac{{y - {y_1}}}{b} = \frac{{z - {z_1}}}{c} \Leftrightarrow \vec r = ({x_1}\hat i + {y_1}\hat j + {z_1}\hat k) + \lambda (a\hat i + b\hat j + c\hat k)\]
NOTE:
Straight lines parallel to coordinate axes:
| | Straight Lines
| Equation |
|---|
| (i) | Through Origin
| y = mx, z = nx
|
| (ii) |
x - axis
|
y = 0, z = 0
|
| (iii) |
y - axis
|
x = 0, z = 0
|
| (iv) |
z - axis
|
x = 0, y = 0
|
| (v) |
Parallel to x - axis
|
y = q, z = r
|
| (vi) |
Parallel to y - axis
|
x = p, z = r
|
| (vii) |
Parallel to z - axis
|
x = p, y = q
|
24. ANGLE BETWEEN A PLANE AND A LINE
25. CONDITION FOR A LINE TO LIE IN A PLANE
(i) Cartesian form: Line \(\dfrac{{x - {x_1}}}{l} = \dfrac{{y - {y_1}}}{m} = \dfrac{{z - {z_1}}}{n}\) would lie in a plane ax + by + cz + d = 0, if ax1 + by1 + cz1 + d = 0 and al + bm + cn = 0.
(ii) Vector form: Line \(\vec r = \vec a + \lambda \vec b\) would lie in the plane \(\vec r \cdot \vec n = d\) if \(\vec b \cdot \vec n = 0\) and \(\vec a \cdot \vec n = d\).
26. COPLANER LINES
27. SKEW LINES (i) The straight lines which are not parallel and non-coplanar i.e. non-intersecting are called skew lines.
If \[\Delta = \left| {\begin{array}{*{20}{c}}{\alpha ' - \alpha }&{\beta ' - \beta }&{\gamma ' - \gamma }\\l&m&n\\{l'}&{m'}&{n'}\end{array}} \right| \ne 0\]
then the lines are skew.
(ii) Vector Form: For lines \({{\vec a}_1} + \lambda {{\vec b}_1}\) and \({{\vec a}_2} + \lambda {{\vec b}_2}\) to be skew, when \(({{\vec b}_1} \times {{\vec b}_2})({{\vec a}_2} - {{\vec a}_1}) \ne 0\) or \([{{\vec b}_1}{{\vec b}_2}({{\vec a}_2} - {{\vec a}_1})] \ne 0\)
(iii) Shortest distance between the two parallel lines \(\vec r = {{\vec a}_1} + \lambda \vec b\) and \(\vec r = {{\vec a}_2} + \mu \vec b\) is \[d = \left| {\frac{{({{\vec a}_2} - {{\vec a}_1}) \times \vec b}}{{|\vec b|}}} \right|\]
28. COPLANARITY OF FOUR POINTS The points A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) are coplaner, then \[\begin{array}{l}\left| {\begin{array}{*{20}{c}}{{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}}\\{{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}}\\{{x_4} - {x_1}}&{{y_4} - {y_1}}&{{z_4} - {z_1}}\end{array}} \right| = 0\\\end{array}\]
29. SIDES OF. PLANE A plane divides the three dimensional space two equal parts. Two points A (x1,y1, z1,) and B (x2, y2, z2) are on the same side of the plane ax + by + cz + d = 0 if ax1 + by1 + cz1 + d and ax2 + by2 + cz2 + d and both positive or both negative and are opposite side of plane if both of these values are in opposite sign.
30. LINE PASSING THROUGH THE GIVEN POINT (x1 y1 z1) AND INTERSECTING BOTH THE LINES (P1 = 0, P2 = 0) AND (P3 = 0, P4 = 0) Get a plane through (x1, y1, z1) and containing the line (P1 = 0, P2 = 0) as P5 = 0
Also get a plane through (x1, y1, z1) and containing the line P3 = 0, P4 = 0 as P6 = 0
Equation of the required line is (P5 = 0, P6 = 0)
31. TO FIND IMAGE OF A POINT W.R.T. A LINE
Let is a given line. Let (x', y', z') is the image of the point P(x1, y1, z1) with respect to the line L.
then,
(i) a(x1 - x') + b(y1 - y') +c(z1 - z') = 0
(ii) \(\dfrac{{\dfrac{{x + {x_1}}}{2} - {x_2}}}{a} = \dfrac{{\dfrac{{{y_1} - y'}}{2} - {y_2}}}{b} = \dfrac{{\dfrac{{{z_1} - z'}}{2} - {z_2}}}{c} = \lambda \)
from (ii) get the value of x', y', z' in terms of λ as x' = 2aλ + 2x2 - x1, y' = 2bλ + 2y2 - y1 and z' = 2cλ + 2z2 - z1. Now, put the values of x', y', z' in (i) get λ and resubstitute the value of λ to get (x', y', z').
Let P(x1, y1, z1) is a given point and ax + by + cz + d = 0 is given plane. Let (x', y', z') is the image point, then
- x' - x1 = λa, y' - y1 = λb, z' - z1 = λc, \( \Rightarrow \) x' = λa + x1, y' = λb + y1, z' = λc + z1,
- \(a\left( {\dfrac{{x' + {x_1}}}{2}} \right) + b\left( {\dfrac{{y' + {y_1}}}{2}} \right) + c\left( {\dfrac{{z' + {z_1}}}{2}} \right) + d = 0\)
from (i) put the values of x', y', z' in (ii) and get the values of λ and re-substitute in (i) to get (x', y', z').
