The Application of Vectors

This word document gives methods to work out angles, intersections and distances between lines, planes and points. Diagrams are included in each case. 

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The Application of Vectors
Items Angle between Distance between Intersection of
Plane and plane 1. 2. 3.
Plane and line 4. 5. 6.
Plane and point 7. 8. 9.
Line and point 10. 11. 12.
Line and line 13. 14. 15.
Point and point 16. 17. 18.
The basics of vectors
A vector is a quantity with both a magnitude and a direction. A scalar is a quantity with just
If I say that I am running at 10ms-1 , this only gives a sense of only how fast I am running ­ i.e. a
scalar. However if I say that I am running at 10ms-1 due north, then it gives a sense of
magnitude and direction ­ i.e. a vector quantity.
To-scale vectors are drawn as arrows, with their length representing magnitude of the
vector quantity and their direction representing the vector's direction.
Vectors are denoted as letters with lines or squiggles above. In textbooks, they are
For some vector a
~ , |~
a| or a denotes the magnitude of the vector.
A vector with a hat denotes a unit vector, which is the respective vector that has unit
magnitude. Hence the unit vector of a
~ is written a
^ (read as `a hat'). a a
^ =a
Vectors can be considered on the 3D Cartesian plane, and can be expressed as the sum of
^, ^
multiples of the unit vectors i ^ , which are parallel to the x , y and z axis respectively.
j and k
The addition and subtraction of vectors is simple: the respective components of vectors are
added or subtracted.

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The position vector of a point in Cartesian space is the fixed vector from the origin to that
point. Hence if A is the point in Cartesian space with coordinates (1, 3, 6) , then its position
^ + 3^
vector OA = i ^.
j + 6 k
^ + c^
Position vectors can be written as column vectors, where (b c d ) = bi ^.
j + dk
Vector multiplication takes two forms:
o Dot (scalar) product: vector times vector = scalar.…read more

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Intersection of two planes
4.…read more

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Distance between a plane and a parallel line
6. Intersection of a plane and a line
7.…read more

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The distance between a plane and a point
9. The intersection of a point and a plane
10. The angle between a line and a point
11.…read more

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The intersection of a line and a point
13. The angle between a line and a line
14.…read more

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The intersection of two non-parallel lines
16. The angle between two points
17. The distance between two points
18.…read more


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