Let and be Euclidean vectors, and the angle between them. Then the dot product of and is denoted and defined as
where for example, denotes the magnitude of .
Since the dot product is an operation on two vectors that returns a scalar value, the dot product is also known as the scalar product.
Geometrically, one can also interpret the dot product as
That is, one can view the dot product as the magnitude of times the magnitude of the component of that points along . is the magnitude of the projection of onto
so the dot product can also be viewed the magnitude of times the magnitude of the component of that points along .
Since and are positive quantities, the sign of the dot product depends on
- If is acute, then is positive, and therefore the dot product is positive.
- If is , then the dot product is zero. Vectors whose dot product vanishes are said to be orthogonal.
- If is obtuse, then the dot product is negative.
Note that since the Euclidean basis unit vectors , , and are mutually perpendicular, it holds that
Given that the magnitude of is and that of is , find when the angle between and is
To find the dot product, we use the formula .
We know and which implies Hence the following answers:
(i) When , and therefore
(ii) When , and therefore
(iii) When , and therefore
If and , what is
We can apply the formula .
We know and . Also, the two vectors are parallel, so and therefore
After we substitute the values in the formula, we get
The dot product has several important and useful properties. Their proofs are fairly straightforward and left as exercises for the reader.
Given find .
The above property tells us that . We are asked to find , which is equivalent to .
Since , it follows that
In Cartesian coordinates, the dot product takes on a convenient form. Suppose that and form angles and , respectively, with the -axis. Recall that the representation in Cartesian coordinates becomes
where , , , and .
In other words, the product of two vectors in Cartesian coordinates is simply the sum of the product of each of the corresponding components of the two vectors. The same applies to vectors in more than two dimensions.
Dot Product in Cartesian Coordinates:
Suppose and . Then
Vectors , and form a triangle, where is the angle between and . Hence, we can apply cosine theorem to this triangle:
Find the cosine of the angle between each of the following pairs of vectors:
a) We have
b) We have
Given two vectors and , for what value of is the angle between them
Then from the definition of dot product, we have
Therefore, for both and , the angle between the two vectors will be .