Vectors

A VECTOR? □Describes the motion of an object □A Vector comprises □Direction □Magnitude □We will consider □Column VectorsColumn Vectors □General VectorsGeneral Vectors □Vector GeometryVector Geometry Size

Column Vectors a Vector a COLUMN Vector 4 RIGHT 2 up NOTE! Label is in BOLD. When handwritten, draw a wavy line under the label i.e.

Column Vectors b Vector b COLUMN Vector? 3 LEFT 2 up

Column Vectors n Vector u COLUMN Vector? 4 LEFT 2 down

Describe these vectors b a c d

Alternative labelling A B C D F E G H

General Vectors A Vector has BOTH a Length & a Direction k can be in any position k k kk All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k

General Vectors k A B C D -k 2k F E Line CD is Parallel to AB CD is TWICE length of AB Line EF is Parallel to AB EF is equal in length to AB EF is opposite direction to AB

Write these Vectors in terms of k k A B C D E F G H 2k 1½k ½k -2k

Combining Column Vectors k A B C D

A B C Simple combinations

Vector Geometry Consider this parallelogram Q O P R a b Opposite sides are Parallel OQ is known as the resultant of a and b

Resultant of Two Vectors □Is the same, no matter which route is followed □Use this to find vectors in geometrical figures

Example Q O P R a b. S S is the Midpoint of PQ. Work out the vector = a + ½b

Alternatively Q O P R a b. S S is the Midpoint of PQ. Work out the vector = a + ½b = b + a - ½b = ½b + a

Example A B C p q M M is the Midpoint of BC Find BC AC= p, AB = q BCBAAC=+ = -q + p = p - q

Example A B C p q M M is the Midpoint of BC Find BM AC= p, AB = q BM ½BC = = ½(p – q)

Example A B C p q M M is the Midpoint of BC Find AM AC= p, AB = q = q + ½(p – q) AM + ½BC = AB = q +½p - ½q = ½q +½p= ½(q + p)= ½(p + q)

Alternatively A B C p q M M is the Midpoint of BC Find AM AC= p, AB = q = p + ½(q – p) AM + ½CB = AC = p +½q - ½p = ½p +½q= ½(p + q)