Three-Dimensional Geometry
Based on the vector algebra discussed in the previous chapter, here are the concepts like, how it can be applied to three-dimensional geometry. Also, the introduction to topics like direction cosines and direction ratios, cartesian and vector equations of a line, and how to find the shortest distance between two lines using these concepts are discussed in this part. Various formulas used in 3-D geometry are,
Distance Formula: The distance between two points A(x1, y1, z1) and B(x2, y2, z2) is given by,
[Tex]AB~=~\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} [/Tex]
While, the distance between two points A(x, y, z) from the origin O(0, 0, 0) is given by,
[Tex]OA~=~\sqrt{x^2+y^2+z^2} [/Tex]
Section Formula: The coordinates of the point R which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) internally or externally in the ratio m:n are given by,
[Tex]\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n},\dfrac{mz_2+nz_1}{m+n}\right)\,\text{and}\left(\dfrac{mx_2-nx_1}{m-n},\dfrac{my_2-ny_1}{m-n},\dfrac{mz_2-nz_1}{m-n}\right) [/Tex]
Mid-Point of the joint of (x1, y1) and (x2, y2) is,
[Tex]\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right) [/Tex]
Coordinates of Centroid of a Triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is,
[Tex]\left(\dfrac{x_1+x_2+x_3}{3},\,\dfrac{y_1+y_2+y_3}{3}\right) [/Tex]
Incentre of triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is,
[Tex]\left(\dfrac{ax_1+bx_2+cx_3}{a+b+c}, \dfrac{ay_1+by_2+cy_3}{a+b+c}, \dfrac{az_1+bz_2+cz_3}{a+b+c}\right) [/Tex]
Centroid of a Tetrahedron with vertices (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), and (x1, y4, z4) is,
[Tex]\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}, \dfrac{z_1+z_2+z_3+z_4}{4}\right) [/Tex]
Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis, and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos α, m = cos β, and n = cos γ. Also, the sum of squares of direction cosines of a line is always 1, i.e.
l2 + m2 + n2 = 1
cos2 α + cos2 β + cos2 γ = 1
Direction Ratios of a Line: Numbers proportional to the direction cosines of a line, are called direction ratios of a line.
- If a, b, and c are direction ratios of a line, then
[Tex]\dfrac{l}{a}~=~\dfrac{m}{b}~=~\dfrac{n}{c} [/Tex]
- If a, b, and care direction ratios of a line, then its direction cosines are
[Tex]l~=~\pm \dfrac{a}{\sqrt{a^2+b^2+c^2}}\\m~=~\pm \dfrac{b}{\sqrt{a^2+b^2+c^2}}\\n~=~\pm \dfrac{c}{\sqrt{a^2+b^2+c^2}} [/Tex]
Angle between two line segments: When a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the acute angle between them, then:
[Tex]\cos\theta~=~\left|\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a^2_1+b^2_1+c^2_1}\sqrt{a^2_2+b^2_2+c^2_2}}\right| [/Tex]
Two lines will be perpendicular if their direction ratios have a relation,
a1a2 + b1b2 + c1c2 = 0
Two lines will be parallel if their direction ratios have a relation,
a1/a2 = b1/b2 = c1/c2
Projection of a line segment on a line: When P(x1, y1, z1) and Q(x2, y2, z2) then the projection of PQ on a line having direction cosines l, m, n is
|l(x2-x1)+m(y2-y1)+n(z2-z1)|
Equation of a Plane
- General form: ax+by+cz+d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.
- Normal form: lx+my+nz = p
- Plane through the point (x1, y1, z1): a(x-x1)+b(y-y1)+c(z-z1) = 0
- Intercept form: [Tex]\dfrac{x}{a}~+~\dfrac{y}{b}~+~\dfrac{z}{c}~=~1 [/Tex]
- Vector form: [Tex] (\vec{r}~-~\vec{a} ).\vec{n}= 0 \text{ or }\vec{r}.\: \vec{n}= \vec{a}.\: \vec{n} [/Tex]
Planes Parallel to Axes
- Plane Parallel to X-axis is by + cz + d = 0
- Plane Parallel to Y-axis is ax + cz + d = 0
- Plane Parallel to Z-axis is ax + by + d = 0
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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