Dot products and Cross products

Dot Products Given any two vectors

and

, we define their dot product as follows: In

and

Remark: The dot product of any two vectors is a scalar i.e. a real number. We can easily derive the following properties of dot product from its definition: Let

be vectors in

and

be a real number. 1.

It turns out that for any two non-zero vectors

and

, it can be shown that their dot product can also be expressed as follows:

when both vectors are non-zero and

is the angle in the range from

formed by translating two non-zero vectors such that their tails meet at a point. In the applet below, bring the two vectors together in the plane to find out their dot product.

Question: Describe what happens to two non-zero vectors and and their dot product when (a) (b) (c) (d) (e)

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Definition: Two vectors are orthogonal if their dot product is zero i.e. if two vectors are orthogonal, they are either both non-zero and perpendicular to each other, or at least one of the vectors is a zero vector. Remark: Zero vector is orthogonal to any vector.

Exercise: (a) Let and . Show that they are orthogonal. (b). Let and . Find the angle between and .

Theorem: Given any two vectors

and

The proof is as follows:

Orthogonal Projections Given non-zero vectors

and

, let

be the line through the tail of

in the direction of

. We define the (orthogonal) projection of

onto

, denoted by

, to be the vector pointing from the tail of

such that its head is the foot of the perpendicular from the head of

to the the line

. In the applet below, the red vector is the projection of

onto

. If

, its magnitude is

and it is in the same direction as

. Hence, we have

. If

, its magnitude is

and it is in opposite direction to

. Hence, we have

(Note: When

, the tail of

is the foot of the perpendicular and hence

.) Combining above, we have the formula for the projection:

Using the dot product, we can also rewrite the formula as follows:

Exercise: Let and . Compute .

Exercise: Given any non-zero vectors and , let i.e. . Show that and are orthogonal. (Decomposition of a vector into its orthogonal components)

An Application - Work A constant force is applied to an object. We define the work done (W) by the force over the displacement of the object as follows:

Therefore, we have

. Example: Given a force

that moves an object from

. Find the work done by

. Therefore, we have

(J stands for "joule", the unit for energy i.e. work done).

Cross Product Given two vectors

and

, we are going to define the cross product of these two vectors. But first of all, we need to recall the definition of the determinant of a matrix: Suppose

is a

matrix. Its determinant

Then we can define the determinant of a

matrix

as follows:

Example:

Write

and

. The cross product,

is defined as follows:

Remark:

Cross product of two vectors in is a vector in .
The order of taking cross product is important: In general,

Example: Given

and

. We have

Properties of Cross Product Recall that if we interchange any two rows of a

matrix, its determinant will be changed as follows:

Therefore, by the definition of cross product, we have the following properties:

and in particular, .
Suppose . (Note: In some textbooks, this is called the scalar triple product). In particular, i.e. is orthogonal to both and .
Let and be real numbers, then .
For any vectors , and . (Distributive laws).

For any two non-parallel vectors

and

, there are exactly two possible directions that are orthogonal to both vectors. We can use the following right-hand rule to determine the direction of the cross product:

In the applet below, you can do the following:

First construct two vectors and by typing "u=vector((1,2,3))" and "v=vector((4,5,6))".
Type "cross(u,v)" to get the cross product .
Draw the plane containing the two vectors through the origin by typing "plane((0,0,0),u,v)". Then you will see that the cross product is perpendicular to the plane.
Verify the right-hand rule using some other vectors.
You can also find , and .

We have the following nice theorem about the norm of a cross product: Theorem:

, where

is the angle between the two vectors. Proof:

(Note: positive square root here because

for

Remark:

is exactly the area of the parallelogram spanned by

and

Exercise: Find the area of the triangle with vertices , and using the above theorem.

Dot products and Cross products

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