Mathematics 1

called a unit vector.

m is a unit vector if = 1.

A unit vector in the direction

of a vector a is given by m =

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Sample problem 2

• Triangles ABC and XYZ are equilateral.
• X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.
• =a ; =b ; = c
• Express each of the

following in terms of a, b

and c.

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Sample problem 2 -answer

• = c
• =-a
• = b+ c ( XCZ) or –a+2c ( AXC)
• =b-a ( ZXB)or 2b-c ( ZBC) or -2a+c ( ABZ)
• =2c

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c
a
b
b
( ZXB)

Vector Components

• a = is a position vector
• of the point P
• ax and ay - the Cartesian
• coordinates of P.
• By Pythagoras' Theorem

=.

• The unit vectors in 2D

corresponding to line

segments of length one

along the positive x and y

axes are denoted by i and j.

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P
O
It follows from the triangle rule that we can write the position vector of the point P with Cartesian coordinates (ax; ay) as
a= axi + ayj

3D Vectors

• The position vector a = of a point

P in 3D with Cartesian coordinates

(x; y; z) is:

• a = = axi + ayj + azk;
• By 3D version of Pythagoras'

Theorem:

=

• Any 3D vector v can be written as a column matrix or a row matrix(a; b; c),and be resolved as:

v = ai + bj + ck; where i = (1; 0; 0), j = (0; 1; 0), k = (0; 0; 1).

Zero vector – with components (0;0;0)

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O
P

Vectors’ addition and scalar multiplication in terms of components

• Let a = (a1; a2; a3), b = (b1; b2; b3) be vectors then
• a + b = (a1; a2; a3)+ (b1; b2; b3) = (a1 + b1; a2 + b2; a3 + b3)
• Let k iz a scalar, then
• ka =(k a1; k b1; k c1) -Multiplying a vector by a scalar will ONLY CHANGE its magnitude. Multiplying a vector by “-1” does not change the magnitude, but it does reverse it's direction
• Example.
• Vectors v and u are given by v = (4 ; 1) and u = (u1 ; u2),
• find components u1 and u2 so that 2 v - 3 u = 0 .
• Solution:
• 2v -3 u =(2*4-3 u1; 2-3 u2)=u1=8/3 ; u1=2/3

Scalar Product

• Multiplying 2 vectors sometimes gives you a SCALAR quantity which we call the SCALAR DOT PRODUCT
• The Dot product formula is given by:
• a.b= cos()
• For parallel vectors in the same

direction (a.b=

• If they are in opposite directions then , cos = -1 and a.b= -
• If a and b are orthogonal, then and cos =0 Hence, a.b = 0:
• Conversely, if a.b = 0 and a and b are non-zero vectors, then cos = =0 and so

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P

Properties of dot product

Summarising we can list the dot product properties:

• u · v = |u||v| cos θ
• dot product with itself is non-negative : u⋅u=|u|2≥0
• dot product is commutative: u · v = v · u
• u · v = 0 when u and v are orthogonal.
• u · v= |u||v| when u and v are parallel
• 0 · 0 = 0
• Scalar Multiplication Property : a(u·v) = (au) · v= u  · (a v)
• Distributive Property : (au + bv) · w = (au) · w + (bv) · w