The Dot Product
Imagine you have two unit vectors, that is, two vectors of unit length:
and
If we multiply their horizontal components together and their vertical components together and then sum we get
This angle is the angle between the two vectors
With this in mind, we define the following operation known as the dot product or scalar product.
If
and
then
This operation is useful because if the angles of inclination of
and are and respectively then
Since the two vectors on the right side of the equation are unit vectors this means
where is the angle between the two vectors
Blue Question: Consider two distinct lines in the xy-plane with slopes
and where . Show that at the point of intersection of the two lines the
angle
they form satisfies
Pink Question: Prove the following facts about the dot product.
a.) The dot product is commutative:
b.) For any scalar
,
c.) For any three vectors, show that the dot product is distributive:
Orange Question: The four hydrogen atoms of a molecule of methane sit at the four vertices of a regular tetrahedron
surrounding a central atom of carbon. The atoms are imagined to be held together by force vectors of equal strength so that the atom is in static equilibrium and the angle between any two force vectors is the same.
Find this angle.
STEP 2013, Math III, #3: The four vertices for of a regular tetrahedron lie on the surface of a sphere with centre at O and of radius 1. The position vector of
with respect to O is for . Use the fact that
to show that
for .
Let
be any point on the surface of the sphere, and let denote the length of the line joining and for .
(i) By writing as , where is the position vector of with respect to O , show that
(ii) Given that
has coordinates and that the coordinates of are of the form , where , show that and , and find the coordinates of
and .
(iii) Show that
By letting the coordinates of
be , show further that
is independent of the position of
.
Orthogonality
An immediate consequence of the definition of the dot product is the following statement
Two vectors are perpendicular, or orthogonal, if and only if their dot product is 0.
Green Question: Suppose is a unit vector. Show that the vector defined by
is orthogonal to .
Skill Set
Vector Operations
No. 1
STEP 2006, Math II, #6: By considering a suitable scalar product, prove that
for any real numbers
, , , , , and . Deduce a necessary and sufficient condition on , , , , , and for the following equation to hold:
.
(i) Show that
for all real numbers , , and .
(ii) Find real numbers
, , and that satisfy both
and .
A complete solution to this problem is posted below.
The Cross Product
Two vectors in space define a set of parallel planes, i.e. any plane that contains those two vectors when they're positioned with their initial points together somewhere in space.
We frequently want to find a vector which is orthogonal to both of these vectors and hence orthogonal to any plane containing the vectors. Therefore we need an operation that produces not a number, but another vector from two original vectors.
We define the cross product in the following way
Written out in full
=
but it's usually easier to use the matrix formulation above.
Let's check that this new vector really is orthogonal to
as promised. You can then verify that it's also orthogonal to .
Keep in mind an important difference between the dot product and the cross product.
The dot product always produces a number from two vectors while the cross product produces a new vector from two vectors.
For this reason, the dot product is often referred to as the scalar product whilst the cross product is referred to as the vector product.
Red Question: Prove that
.
Aqua Question: Prove that both the dot product and cross product are distributive. In other words, show that
i)
ii)
Yellow Question: Show that if two vectors are parallel then their cross product is the zero vector.
Area
The next part is difficult to do without messy components, so we will dive right in. Consider the following calculation:
So we have the statement
We can derive one more useful fact from this by inserting the cosine formula from the dot product.
where
is the angle between the two vectors, . Then
So we can summarize the dot and cross products by saying that if
is the angle between the two vectors, , then
and
This last result gives as an elegant way to calculate area.
Consider triangle
at right formed by two vectors and and
the segment that connects their endpoints.
A little right triangle trig should convince you that the height of the triangle is given by
therefore the area of the triangle is given by
Example: Find the area of the triangle whose vertices are given by the three points and and
These three points are graphed at left. We create the two vectors
and
Then
So the area of the triangle is
Skill Set
Vector Operations
No. 2
Purple Question: Suppose that you have two non-parallel vectors
and and a vector such that is a linear combination of and .
Prove that
. What does this result say about how we can determine if any three vectors are coplanar?
[Note: An object
is a linear combination of two others and if and only if for some real numbers and .]
Brown Question: Consider any three non-coplanar vectors
, and . Explain the geometrical significance of the quantity
Violet Question: Given two vectors
and , find a unit vector that bisects the angle formed by and .
STEP 2011, Math II, #5: The points A and B have position vectors
with respect to an origin O, and O, A and B are non-collinear. The point C, with position vector
and
, is the reflection of B in the line through O and A. Show that can be written in the form
where
The point D, with position vector
, is the reflection of C in the line through O and B. Show that can be written in the form
for some scalar
to be determined.
Given that A, B, and D are collinear, find the relationship between
and . In the case , determine the cosine of AOB and describe the relative position of A, B, and D.
Supplement: