Tutorials in Vectors: At a glance
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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 lineParametric 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 lineParametric Vectorial equation of a line through two given points with position vectors a & b is r = a + t(ba)
Condition for collinearity of three pointsCondition 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 planeParametric 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 + pc 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(ba) + pc
We will also discuss :
Condition for coplanarity of four pointsCondition 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 planeThe 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 (ra).n = 0. Plane coaxal 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 linesAngle 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 planesAngle between a line and a plane.r.n1 = q1, r.n2 = q2 isθ = cos1 (n1).(n2)n1n2 Equation of a planewhich 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 noncollinear points A, B, C with position vector a, b, c is also given by (ra)x[(ba)x(ca)] = 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 = (ca).(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 = 0We 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. Coplanar 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.
 

