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http://www.intmath.com/vectors/5-dot-product-vectors-2-dimensions.php

5. Dot Product (aka Scalar Product) in 2 Dimensions

If we have any 2 vectors P and Q, the dot product of P andQ is given by:

P Q = |P| |Q| cos

where

|P| and |Q| are the magnitudes of P and Q respectively, and

is the angle between the 2 vectors.

The dot product of the vectors P and Q is also known as the scalar product since it alwaysreturns a scalar value.

The term dot product is used here because of the notation used and because the term"scalar product" is too similar to the term "scalar multiplication" that we learned aboutearlier.

Examples

1. Find the dot product of the force vectors F1= 4 N and F2= 6 N acting at 40 to each other

as in the diagram.

Answer

The dot product is:

F1 F2= |F1| |F2| cos

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= (4)(6) cos 40

= 18.38 N

2. Find the dot product of the vectors P and Q if |P| = 7 units and |Q| = 5 units and they areacting at right angles to each other.

Answer

The required dot product is:

P Q= |P| |Q| cos

= 7 5 cos 90

= 0

The second example illustrates an important point about how scalar products can be used

to find out if vectors are acting at right angles, as follows.

Dot Product and Perpendicular Vectors

If 2 vectors act perpendicular to each other, the dot product (ie scalar product) of the 2vectors has value zero.

This is a useful result when we want to check if 2 vectors are actually acting at right angles.

Dot Products of Unit Vectors

For the unit vectors i (acting in thex-direction) andj (acting in they-direction), we have thefollowing dot (ie scalar) products (since they are perpendicular to each other):

i j = j i = 0

Exercise

What is the value of these 2 dot products:

(a) i i

(b)j j

Answer

(a)

i i= | i | | i | cos 0

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= 1 1 1

= 1

(b)

j j= | j | | j | cos 0= 1 1 1

= 1

Alternative Form of the Dot Product

Recall that vectors can be written using scalar products of unit vectors.

If we have 2 vectors P and Q defined as:

P = ai + bj

Q = ci + dj,

where

a, b, c, dare constants;

i is the unit vector in thex-direction; and

j is the unit vector in the y-direction,

then it can be shown that the dot product (scalar product) of P and Q is given by:

P Q = ac + bd

Answer

For convenience, let vectors P and Q be as shown on the graph. (P is horizontal.)

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Draw the altitude from vector P to the terminal point (c, d) of Q.

Simple trigonometry gives us:

cos =c / | Q |, so

c = | Q| cos

and

sin =d/ | Q |, so

d= | Q | sin

Since P is horizontal,

a = | P |

b = 0

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So

ac + bd

= a | Q | cos + 0

= | P | | Q | cos

Therefore

P Q = | P | | Q | cos =ac + bd

This result can be generalized for P and Q in any orientation.

Example - Alternative Form of the Dot Product

Find P Q if

P = 6 i + 5j and

Q = 2 i 8j

Answer

Now we see another use for the dot product finding the angle between vectors.

Angle Between Two Vectors

We can use the dot product to find the angle between 2 vectors. For the vectors P andQ,the dot product is given by

P Q = |P| |Q| cos

Rearranging this formula we obtain the cosine of the angle between P and Q:

To find the angle, we just find the inverse cosine of the expression on the right.

So the angle between 2 vectors P and Q is given by

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Example

Find the angle between the vectors P = 3 i 5j and Q = 4 i + 6j.

Answer