The equation of a straight line vector passing through point and is parallel to vector is where is the position vector of a general point on the line.
So by taking varying values of you can find the position vectors of varying points that lie on the straight line.
The line has cartesian equation Colinear: Points are said to be collinear if they all lie on the same straight line
Plane Vectors
The vector of a plane that through position vector is: where r is the position vector of a general point on the plane with being the position vector of a fixed arbitrary point on the plane and are non-parallel, non-zero vectors on the planeThe intuition of this is that gets you from the origin onto the plane and allow you to move across the plane
The cartesian equation of a plane is where the normal vector to the plane is . You can find the cartesian equation given the vector equation using the cross product (only for needed FP1)
Dot Product
The dot product (also called scalar product) between 2 vectors is given by the equation where is the acute angle between and (when they are pointing away from their intersection).
The scalar product is Commutative
The dot product is also Distributive meaning that you can compute the scalar product without
Using this you can compute the angle with just a simple expression
Planes in The Form
Suppose a plane passes through and has normal vector and let be an arbitrary point on . Then . Since lies on , so where is a constant so . This can also be written as . Where represents the perpendicular distance from the origin to the place if
Angles Between Vectors and Planes
The Acute angle between 2 intersecting Straight Lines(with direction vectors ) is given with the formula
The Acute angle between Straight Line(with direction vectors ) and Plane(with equation ) is given with the formula
The Acute angle between 2 intersecting Planes(with equations vectors and ) is given with the formula
Points of Intersection
To find if 2 lines and intersect set them equal and solve the system of equations, if solutions for and exist then they intersect, if not then there is no intersection.
Two lines are skew if they do not intersect but are parallel meaning solutions for and don’t exist but their direction vectors are scalar multiples of each other.
To find the intersection of a plane and a line have the plane in the form then sub in the vector equation into the plane equation.
Perpendicular Distances
To find the shortest/perpendicular distance between 2 lines and , Let be a point on and be a point on then shortest/perpendicular distance between and will be the straight line segment that is perpendicular to both lines. So you can then solve for and
To find shortest distance between a line and a point , Let be a point on then . You can then find the coordinates of so you can find the distance
The perpendicular distance from the origin to the plane is simply k
The perpendicular distance from a point to the plane is
The distance between two parallel planes and is
A-Level Further Pure 1
Vector Cross Product
Definition
The cross product is defined as .
It is Commutative and Distributive
is the angle between and where . And is the unit vector perpendicular to both and
Right Hand Rule
The direction of depends on the right-hand rule
Swapping and means that therefore
Areas
Finding Volumes - Scalar Triple Product
Straight Lines
Suppose is the position vector of a point on a line and the line is parallel to . Let be the position vector of a general point on the line. so . Since is parallel to , so
Direction of Cosines
If a line is parallel to the vector the direction ratios are and the direction of cosines of the line are
The sum of the squares of the direction cosines is always one
