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Inquiry	in	DP	Mathematics	Structured	Jennifer	Wathall	 Page	1	
	
Scalar	Products	
Here	is	an	example	for	a	pair	of	
vectors	
	
	
	
																																																																																										
5
1
.
3
3
	gives	a	product	of	18	
Why	do	you	think	this	is	called	the	scalar	product?		
	 	
Vectors	all	have	two	basic	properties:	
	
	A	systematic	exploration	of	the	scalar	product	should	look	at	each	of	these	in	turn.	
First	draw	pairs	of	vectors	where	the	directions	stay	the	same,	but	the	magnitude	changes.	This	will	
involve	keeping	the	angle	between	your	pairs	constant	but	changing	the	length.		
	
	
For	the	vectors	!
𝒂 𝟏
𝒂 𝟐
! 𝒂𝒏𝒅 !
𝒃 𝟏
𝒃 𝟐
! 	
The	Scalar	Product	is	defined	as	a1b1	+	a2b2	
	
Can	you	find	a	relationship	between	the	
magnitude	and	the	scalar	product?
Inquiry	in	DP	Mathematics	Structured	Jennifer	Wathall	 Page	2	
	
Now	try	keeping	the	magnitude	the	same,	but	changing	the	direction.	Try	keeping	one	vector	
constant	but	rotating	the	second	one	to	see	what	happens	to	the	scalar	product.	
	
	
	
Can	you	sketch	a	graph	of	how	the	product	varies	as	the	angle	varies	from	00	to	3600?	
	
	
	
	
	
	
What	does	the	graph	look	like?	
	
	
	
	
	
When	is	it	positive,	and	when	negative?		
	
When	is	it	zero?	
	
	
When	is	it	at	its	largest	and	when	is	it	at	
its	smallest?	
	
What	generalisations	can	you	make	about	the	scalar	product	and	the	magnitude	
and	the	direction	of	two	vectors?
Inquiry	in	DP	Mathematics	Structured	Jennifer	Wathall	 Page	3	
	
Extension		
	
Find	the	length	c	in	two	
different	ways.		One	way	
should	use	the	angle	θ,	
and	one	should	not.	Now	
find	a	formula	for	the	
scalar	product	of	two	
unit	vectors.

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Scalar products structured