Dot products and Cross products
Dot Products
Given any two vectors and in or , we define their dot product as follows:
In , and ,
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 or and be a real number.
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It turns out that for any two non-zero vectors and in or , 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 to 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)
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 in or ,
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
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