Mechanics

calculus-based physics for scientists and engineers

Vector Multiplication

The Dot Product

While adding vectors can still seem like traditional addition with different notation, vector multiplication is nothing like traditional multiplication all. In fact, maybe calling these operations vector multiplication is a bad idea. Well, it's too late now. The first thing to notice is that there are different kinds of vector multiplication. Each has three different names. The first one can be called the dot product (because the notation is a dot), the scalar product (because the result is a scalar), or the inner product (this won't make any sense until a more advanced math class).

The dot product is telling you how large two vectors are and to what degree they are aligned. It is the product of the magnitude of one vector multiplied by the projection of the other vector on that vector. Two ways to calculate the dot product are: $$ \vec{A} \cdot \vec{B} = AB\cos \theta $$ where \(A\) and \(B\) are the magnitudes and \(\theta\) is the angle between the vectors when placed tail to tail, and $$ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z $$ is the dot product given in terms of the components.

The dot product is commutative. $$ \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} $$ Multiplication of a dot product by the scalar \(c\) can be calculated by: $$ c (\vec{A} \cdot \vec{B}) = (c \vec{A}) \cdot \vec{B} = \vec{A} \cdot (c \vec{B}) $$

The Cross Product

The second type of vector multiplication is called the cross product (because the notation is a cross), the vector product (because the result is a vector), or the outer product (yeah, that still won't make sense).

The cross product is telling you how large two vectors are and to what degree they are perpendicular. The result is also a vector that points perpendicular to the plane containing the original two vectors. Because there are two such directions, the correct direction is given by the right hand rule (shown in the video). Two ways to calculate the cross product are: $$ \vec{A} \times \vec{B} = AB\sin \theta\, \hat{n} $$ where \(A\) and \(B\) are the magnitudes, \(\theta\) is the angle between the vectors when placed tail to tail, and \(\hat{n}\) is a unit vector whose direction is given by the right hand rule in the video. If you have the components, you can calculate the component form of the cross product directly: $$ \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\, \hat{i} + (A_z B_x - A_x B_z )\, \hat{j} + (A_x B_y - A_y B_x)\, \hat{k}$$

The cross product is not commutative. $$ \vec{A} \times \vec{B} = - \vec{B} \times \vec{A} $$ Multiplication of a cross product by the scalar \(c\) can be calculated by: $$ c (\vec{A} \times \vec{B}) = (c \vec{A}) \times \vec{B} = \vec{A} \times (c \vec{B}) $$

Exercises

Do Now!   Do these exercises immediately.

Not Now!   Do these after you start to forget the topic, say in a week.

More!   More exercises if you want. Maybe review before a test.

Required Problems

Solution

1. Show that by using the dot product on three vectors comprising three sides of a triangle, you can deduce the law of cosines.

Optional Problems