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Applying Vectors to Geometric Problems
Vectors and Geometry
Parametric Vectorial equation of a line
Parametric Vectorial equation of a lineabra b
Parametric Vectorial equation of a lineParametric Vectorial equation of a line a br a(b-a)
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,x, y, z,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 abcr abc
a & b cr = a + (b-a) +c
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,
Normal form of vector equation of a planeThe equation r.nnormal to the vector, n. q 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
Plane co-axal with given planes
Equation of a plane passing through the line of intersection of two given planes r.n1 r.n2 (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)ijkijk cosθ = l1 l2+ m1 m2+ n1 n2
Angle between two planesAngle between a line and a plane.r.n1, r.n2 isθ = cos-1 (n1).(n2)|n1||n2| 
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]
r.[bxc + cxa] = [a b c]
(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 = 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
In case you're interested in learning more about Vectors, here's the full set of tutorials we have :
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