Component rules
Consider two vectors:
a = x1i + y1j + z1k and b = x2 i + y2 j + z2k
in three dimensional space.
a = b implies that x1= x2 y1= y2 z1= z2
a + b = ( x1+ x2 )i + ( y1+ y2 )j + ( z1+ z2)k
a - b = ( x1- x2 )i + ( y1- y2 )j + ( z1- z2)k
ma = mx1i + m y1j + mz1k where m is a scalar quantity
in 3D space, if point A has position vector a and point B has position vector b then the distance AB is given by:
Equation straight line - single point & parallel vector given
A(x1 y1 z1) is a fixed point on the line
a is the position vector for point A a = x1i + y1 j+ z1k
s is a vector parallel to the line s = l i + m j+ n k
( l m n are called the direction ratios of the line)
r is the position vector for an arbitrary point P(x, y, z) on the line.
Equation of a straight line - two points given
A(x1 y1 z1) is a fixed point on the line
a is the position vector for point A a = x1i + y1 j+ z1k
B(x2 y2 z2) is a fixed point on the line
b is the position vector for point B b = x2i + y2 j+ z2k
r is the position vector for an arbitrary point P(x, y, z) on the line.
