On…

a.k.aDot and Cross products of vectors

Let’s cut to the chase………..�

What is a

is also known as

.

� If we have two vectors; “a” and “b” ,then the scalar or the dot product of the two vectors is represented as such;

�When a=<a1,a2,a3> and b=<b1,b2,b3> that is a and b are 3-D vectors.

�a.b = <a1b1+a2b2+a3b3>

�Note : the dot product of two vectors is always a scalar; the reason for the name - scalar product.

So think about it this way…

� a DOT

…looks kinda like a RING; a wedding ring

ONE

Example

�Find a.b where a =2i-4j+3k and b=i+5j-2k�Solution� a.b=[(2)(1)+(-4)(5)+(3)(-2)]

=2 - 20 - 6=-24

therefore a.b= -24 (a single value – a scalar)

Properties of Dot products of Vectors

�1) a.a= � 2

�2)a.b=b.a(commutativity)

�3)a.(b+c)=a.b+a.c(right distributivity)

�4)(ca).b=c(a.b)=a.(cb)

�5)0.a=0

How on earth can a.a =|a|2 ?� From the left hand side

� Since a= <a1, a2, a3>

� a.a = <a1, a2, a3> . <a1, a2, a3>

� a.a = <a1.a1, a2.a2, a3.a3>

� a.a = a12 +a2

2 + a32

� Conversely, starting from the right hand side we have

� |a| = √a12 +a2

2 + a32

� |a|2 =a12 +a2

2 + a32

Orthogonal Vectors� Two vectors a and b are said to be orthogonal if their scalar product is

equal to zero.

� a.b=0 this happens when the vectors are perpendicular to each other.

� Note : where � is the angle between the two vectors dot product yields the following results;

� a.b=0 when � =900

� a.b>0 when �< 900

� a.b<0 when �> 900

Vector Product

can also be referred to as of vectors.

�Unlike scalar product ,cross product of two

vectors results in another vector.

So think about it this way…� a CROSS

Is a shape made up from

…and when vectors are crossed together, they give another vector ( in terms of i,j and k) and not a single value (scalar)

� If I have the vectors a and b then; b=b1i+b2j+b3k or <b1, b2, b3>

� Where a=a1i+a2j+a3k or <a1, a2, a3>

� Then “a cross b” a x b

= a1i ((((b1i+b2j+b3k)+)+)+)+a2j ((((b1i+b2j+b3k)+)+)+)+a3k ((((b1i+b2j+b3k))))

OROROROR

= a= a= a= a1111 <b1, b2, b3> a2<b1, b2, b3> a3<b1, b2, b3>

It’s just like regular multiplication or expansion... HOW?

� For instance, if I am to multiply (x2+x+2) and (2x2 + 4x2 – 5)

� (x2+x+2) x (2x2 + 4x2 – 5)

� X2 (2x2 + 4x2 – 5) + x (2x2 + 4x2 – 5) + 2 (2x2 + 4x2 – 5)

� aaaa1111 <b1, b2, b3> a2<b1, b2, b3> a3<b1, b2, b3>

� Remember: Each term will have an i,j or k attached to it. And when any of these letters multiply each other they’ll give the following….

Compare

The cross product cycle

� clockwise movement

will yield a positive product.

i

jk

i

jk

anticlockwise movement will

yield a negative product.

+i

+j +k

-i

-k-j

�we also should note the following;�i x j= k j x i= -k

�j x k= i k x j= -i

�k x i= j i x k= -j

�i x i = 0 j x j = 0 k x k = 0

2ND METHOD�

ADDITIONAL INFO� The lengths of dot and cross products are given by the following

� DOT PRODUCT

� |a.b | = |a | |b | cos ѳ

� CROSS PRODUCT

� |a x b | = |a | |b | sinѳ

� SIN (sine) SIN (sine) SIN (sine) SIN (sine) has been paid for on the CROSSCROSSCROSSCROSS

Hope you guys got most, and hopefully, all

of these…

�If you did or didn’t, let’s try these ones out

TRY THEM

�If a= <2,-3,5> and b = <-3,1,6>

�or if you like- a= (2i-3j+5k) and b= (-3i+j+6k)

�Evaluate

�i) a . b

�ii) a x b

� May God perfect your understanding of this in Jesus’ name.