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Applying Vectors to Geometric Problems - Parametric Vectorial equation of a line and Plane, Condition for collinearity of three points, Shortest distance between two lines, Perpendicular distance of a point from a plane or line, Angles between lines and planes






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                                                                            Applying Vectors to Geometric Problems


A quick outline of the topics covered in this tutorial :

Vectors and Geometry


Parametric Vectorial equation of a line

Parametric Vectorial equation of a line through a point with a position vector a and is parallel to a vector, b, is r = a + tb, where r is the position vector of any point, P, on the line. Each point P on the line arises for different values of scalar t.

Parametric Vectorial equation of a line

Parametric Vectorial equation of a line through two given points with position vectors a & b is r = a + t(b-a)

Condition for collinearity of three points

Condition for collinearity of three points : The necessary and sufficient condition for three points A, B, C with position vectors a, b, c, respectively to be collinear is that there exist three scalars x, y, z, not all zero, such that xa + yb + zc = 0, x + y + z = 0

Parametric Vectorial equation of a plane

Parametric Vectorial equation of a plane which passes through the point with position vector a, and which is parallel to the vectors b and c is r = a + tb + p  where t and p are scalar parameters.
And, Parametric Vectorial equation of a plane through two given points with position vectors a & b and parallel to vector c is r = a + t(b-a) + pc

We will also discuss :

Condition for coplanarity of four points

Condition for coplanarity of four points : The necessary and 
sufficient condition for four points A, B, C, D with position vectors a, 
b, c, d respectively to be coplanar is that there exist four scalars x,y, z, t not all zero, such that  xa + yb + zc + td = 0, x + y + z + t = 0

Normal form of vector equation of a plane

The equation r.n = q is the equation of a plane which is 
normal to the vector, n.  q= is the length of perpendicular 
from origin of reference to the plane.
Equation of a plane which is normal to the vector, n, and passing through a point A, with position vector a is (r-a).n = 0.


Plane co-axal with given planes

Equation of a plane passing through the line of intersection of two given planes r.n1 = q1, r.n2 = q2 and passing through a point with position vector a is
(r.n1 – q1). (a.n2– q2) = (r.n2– q2). (a.n1 – q1)

Angle between two lines

Angle between two lines : Let l1, m1, n1; l2, m2, n2 be the direction cosines of two given lines. The vectors of unit length along given (l1)i + (m1)j + (n1)k and (l2)i + (m2)j + (n2)k
cosθ = l1 l2+ m1 m2+ n1 n2

Angle between two planes

Angle between a line and a plane.
r.n1 = q1, r.n2 = q2 is

θ = cos-1 (n1).(n2)|n1||n2|




Equation of a plane

which passes through the point with position vector a, and which is parallel to the vectors b and c is also given by [r b c] = [a b c]


through two given points A, B, with position vector a, b and parallel to given vector c is also given by r.[bxc + cxa] = [a b c]


through three given non-collinear points A, B, C with position vector a, b, c is also given by (r-a)x[(b-a)x(c-a)] = 0


Coplanarity of two lines :

r = a + tb, r = c + pd are coplanar if

[c  b  d] = [a  b  d]


Shortest distance between two lines :

r = a + tb, r = c + pd is


LM = |(c-a).(bxd)||bxd| = |[c b d]- [a b d]||bxd|


Cartesian Equations :

Normal form of Cartesian equation of a plane (r.n = q), where r=xi+yj+zk, n = ai + bj + ck is ax + by + cz = p or ax + by + cz + d = 0

We will also discuss how to find : Equation of a plane passing through three points P1(x1, y1, z1), P2(x2, y2, z2), P3(x3, y3, z3), Intercept form of equation of a plane which makes intercepts a, b, c on x, y, z axes, Equation of a line which passes through a point A(x1, y1, z1) having direction ratios p, q, r, Equation of a line which passes through two points P1(x1, y1, z1) and P2(x2, y2, z2), Coplanarity of two lines .





Perpendicular distance of a point (A, with position vector a) from a plane
Planes bisecting the angles between two given planes
Perpendicular distance of a point (A, with position vector a) from a line (r = b + ct)
Coplanarity of two lines
Shortest distance between two lines

... and other cases where vectors are used to measure geometric properties


Tutorial with Solved Problems :









In case you're interested in learning more about Vectors, here's the full set of tutorials we have :

Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)
Introducing a vector, position vectors, direction cosines, different types of vectors, addition and subtraction of vectors. Vector and Scalar products. Scalar Triple product and Vector triple product and their properties. Components and projections of vectors.

Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )
Solved examples and problem sets based on the above concepts. 



Vectors 2a ( Theory and Definitions: Vectors and Geometry ) Vectors and geometry. Parametric vectorial equations of lines and planes. Angles between lines and planes. Co-planar and collinear points. Cartesian equations for lines and planes in 3D. 

Vectors 2b ( Solved Problem Sets: Vectors and Geometry )

Solved examples and problem sets based on the above concepts.






Vectors 3a ( Theory and Definitions: Vector Differential and Integral Calculus ) Vector Differential Calculus. Derivative, curves, tangential vectors, vector functions, gradient, directional derivative, divergence and curl of a vector function; important formulas related to div, curl and grad. Vector Integral Calculus. Line integral, independence of path, Green's theorem, divergence theorem of Gauss, green's formulas, Stoke's theorems.


Vectors 3b ( Solved Problem Sets: Vector Differential and Integral Calculus ) - Solved examples and problem sets based on the above concepts.



 

 

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