Notes : VECTORS

VECTORS

A vector describes a movement from one point to another.

Vector notation

A vector quantity has both direction and magnitude (size).

(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)

For example this arrow represents a vector. The direction is given by the arrow, while the length of the line represents the magnitude.

This vector can be written as:  , a, or  .

In print, a is written in bold type. In handwriting, the vector is indicated by putting a squiggle underneath the letter: 

Vector 'arithmetic'

Equal vectors

If two vectors have the same magnitude and direction, then they are equal.

Adding vectors

Look at the graph below to see the movements between PQ, QR and PR.

Vector  followed by vector  represents a movement from P to R.

Written out the vector addition looks like this

Subtracting vectors

Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).

Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors  and ?

You could say it is vector  followed by a backwards movement along .

So we can write the path from X to Z as

Written out in numbers it looks like this:

Question

If x =  , y =  and z =  , find:

1. -y
2. x - y
3. 2x + 3z

Resultant vectors

To travel from X to Z, it is possible to move along vector  followed by . It is also possible to go directly along .

 is therefore known as the resultant of and  .

VECTORS IN 3D SPACE

SCALAR PRODUCT 

The scalar product, also called the dot product, of two vectors is a number ( scalar quantity) obtained by performing a specific operation on the vector components. The dot product has meaning only for pairs of vectors having the same number of dimensions. The symbol for dot product is a heavy dot ( ).

Scalar product is used to prove whether the two vectors are parallel or perpendicular. it is also used to determine the angle between two vectors. 

CROSS PRODUCT

The cross product, also called the vector product, is an operation on two vectors. The cross product of two vectors produces a third vector which is perpendicular to the plane in which the first two lie.

LINES IN SPACE

* r = a + tu ;

   where r represents position, a represents a point and u represents direction.

PLANE

The angle between a line, r, and a plane, π, is the angle between r and its orthogonal projection onto π, r'.The formula used to find out the equation of plane is r.n = a.n.