3.4: Vector Product (Cross Product)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let →A and →B be two vectors. Because any two non-parallel vectors form a plane, we denote the angle θ to be the angle between the vectors →A and →B as shown in Figure 3.27. The magnitude of the vector product →A×→B of the vectors →A and →B is defined to be product of the magnitude of the vectors →A and →B with the sine of the angle θ between the two vectors,
|→A×→B|=|→A||→B|sin(θ)
The angle θ between the vectors is limited to the values 0≤θ≤π ensuring that sin(θ) ≥ 0.
The direction of the vector product is defined as follows. The vectors →A and →B form a plane. Consider the direction perpendicular to this plane. There are two possibilities: we shall choose one of these two (the one shown in Figure 3.27) for the direction of the vector product →A×→B using a convention that is commonly called the “right-hand rule”.
Right-hand Rule for the Direction of Vector Product
The first step is to redraw the vectors →A and →B so that the tails are touching. Then draw an arc starting from the vector →A and finishing on the vector →B . Curl your right fingers the same way as the arc. Your right thumb points in the direction of the vector product →A×→B (Figure 3.28).
You should remember that the direction of the vector product →A×→B is perpendicular to the plane formed by →A and →B. We can give a geometric interpretation to the magnitude of the vector product by writing the magnitude as |→A×→B|=|→A|(|→B|sin(θ)). The term |→A|sinθ is the projection of the vector →A in the direction perpendicular to the vector →B as shown in Figure 3.29(b). The vector product of two vectors that are parallel (or anti-parallel) to each other is zero because the angle between the vectors is 0 (or π) and sin(0) = 0 (or sin(π) = 0). Geometrically, two parallel vectors do not have a unique component perpendicular to their common direction
Properties of the Vector Product
- The vector product is anti-commutative because changing the order of the vectors changes the direction of the vector product by the right hand rule: →A×→B=−→B×→A
- The vector product between a vector c→A where c is a scalar and a vector →B is c→A×→B=c(→A×→B) Similarly, →A×c→B=c(→A×→B).
- The vector product between the sum of two vectors →A and →B with a vector →C is (→A+→B)×→C=→A×→C+→B×→C Similarly, →A×(→B+→C)=→A×→B+→A×→C.
Vector Decomposition and the Vector Product: Cartesian Coordinates
We first calculate that the magnitude of vector product of the unit vectors →i and →j:
|ˆi׈j|=|ˆi‖ˆj|sin(π/2)=1
because the unit vectors have magnitude |ˆi|=|ˆj|=1 and sin(π/2)=1. By the right hand rule, the direction of
→i×→j
is in the +ˆk as shown in Figure 3.30. Thus ˆi׈j=ˆk
We note that the same rule applies for the unit vectors in the y and z directions, ˆj׈k=ˆi,ˆk׈i=ˆj By the anti-commutatively property (1) of the vector product, ˆj׈i=−ˆk,ˆi׈k=−ˆj The vector product of the unit vector ˆi with itself is zero because the two unit vectors are parallel to each other, ( sin(0) = 0 ), |ˆi׈i|=|ˆi‖ˆi|sin(0)=0.The vector product of the unit vector ˆj with itself and the unit vector ˆk with itself are also zero for the same reason, |ˆj׈j|=0,|ˆk׈k|=0.
With these properties in mind we can now develop an algebraic expression for the vector product in terms of components. Let’s choose a Cartesian coordinate system with the vector →B pointing along the positive \(x\)-axis with positive \(x\)-component Bx. Then the vectors →A and →B can be written as
→A=Axˆi+Ayˆj+Azˆk\\overrightarrow{\mathbf{B}}=B_{x} \hat{\mathbf{i}} \nonumber \]respectively. The vector product in vector components is→A×→B=(Axˆi+Ayˆj+Azˆk)×Bxˆi
This becomes,
→A×→B=(Axˆi×Bxˆi)+(Ayˆj×Bxˆi)+(Azˆk×Bxˆi)=AxBx(ˆi׈i)+AyBx(ˆj׈i)+AzBx(ˆk׈i) =−AyBxˆk+AzBxˆj
The vector component expression for the vector product easily generalizes for arbitrary vectors
→A=Axˆi+Ayˆj+Azˆk→B=Bxˆi+Byˆj+Bzˆk
to yield
→A×→B=(AyBz−AzBy)ˆi+(AzBx−AxBz)ˆj+(AxBy−AyBx)ˆk
Vector Decomposition and the Vector Product: Cylindrical Coordinates
Recall the cylindrical coordinate system, which we show in Figure 3.31. We have chosen two directions, radial and tangential in the plane, and a perpendicular direction to the plane.
The unit vectors are at right angles to each other and so using the right hand rule, the vector product of the unit vectors are given by the relations
ˆr׈θ=ˆkˆθ׈k=ˆrˆk׈r=ˆθ Because the vector product satisfies →A×→B=−→B×→A, we also have that ˆθ׈r=−ˆkˆk׈θ=−ˆrˆr׈k=−ˆθFinally ˆr׈r=ˆθ׈θ=ˆk׈k=→0
Example 3.6: Vector Products
Given two vectors, →A=2ˆi+−3ˆj+7ˆk and →B=5ˆi+ˆj+2ˆk, find →A×→B
Solution
→A×→B=(AyBz−AzBy)ˆi+(AzBx−AxBz)ˆj+(AxBy−AyBx)ˆk=((−3)(2)−(7)(1))ˆi+((7)(5)−(2)(2))ˆj+((2)(1)−(−3)(5))ˆk=−13ˆi+31ˆj+17ˆk
Example 3.7: Law of Sines
For the triangle shown in Figure 3.32(a), prove the law of wines, |→A|/sinα=|→B|/sinβ=|→C|/sinγ, using the vector product.
Figure 3.32(b): Vector analysis. (CC BY-NC; Ümit Kaya)
Solution
Consider the area of a triangle formed by three vectors →A, →B, and →C, where →A+→B+→C=0 (Figure 3.32(b)). Because →A+→B+→C=0, we have that 0=→A×(→A+→B+→C). Thus →A×→B=−→A×→C or |→A×→B|=|→A×→C|.
From Figure 17.7b, we see that
|→A×→B|=|→A||→B|sinγ
and
|→A×→C|=|→A||→C|sinβ.
Therefore,
|→A||→B|sinγ=|→A||→C|sinβ,
and hence
|→B|/sinβ=|→C|/sinγ.
A similar argument shows that |→B|/sinβ=|→A|/sinα proving the law of sines.
Example 3.8: Unit Normal
Find a unit vector perpendicular to →A=ˆi+ˆj−ˆk and →B=−2ˆi−ˆj+3ˆk.
Solution
The vector product →A×→B is perpendicular to both →A and →B. Therefore the unit vectors ˆn=±→A×→B/|→A×→B are perpendicular to both →A and →B. We first calculate
→A×→B=(AyBz−AzBy)ˆi+(AzBx−AxBz)ˆj+(AxBy−AyBx)ˆk=((1)(3)−(−1)(−1))ˆi+((−1)(2)−(1)(3))ˆj+((1)(−1)−(1)(2))ˆk=2ˆi−5ˆj−3ˆk
We now calculate the magnitude |→A×→B|=(22+52+32)1/2=(38)1/2.Therefore the perpendicular unit vectors areˆn=±→A×→B/|→A×→B|=±(2ˆi−5ˆj−3ˆk)/(38)1/2
Example 3.9: Volume of Parallelepiped
Show that the volume of a parallelepiped with edges formed by the vectors →A, →B, and →C is given by →A⋅(→B×→C)
Solution
The volume of a parallelepiped is given by area of the base times height. If the base is formed by the vectors →B,and→C, then the area of the base is given by the magnitude of →B×→C. The vector →B×→C=|→B×→C|ˆn where ˆn is a unit vector perpendicular to the base (Figure 3.33).
The projection of the vector →A along the direction ˆn gives the height of the parallelepiped. This projection is given by taking the dot product of →A with a unit vector and is equal to →A⋅ˆn=height. Therefore →A⋅(→B×→C)=→A⋅(|→B×→C|)ˆn=(|→B×→C|)→A⋅ˆn=(area)(height)=(volume)
Example 3.10: Vector Decomposition
Let →A be an arbitrary vector and let →n be a unit vector in some fixed direction. Show that →A=(→A⋅ˆn)ˆn+(ˆn×→A)׈n.
Solution
Let →A=A‖ˆn+A⊥ˆe where A‖ is the component →A in the direction of ˆn, ˆe is the direction of projection of →A in a plane perpendicular to ˆn, and A⊥ is the component of →A in the direction of ˆe. Because ˆe⋅ˆn=0 we have that →A⋅ˆn=A‖. Note that
ˆn×→A=ˆn×(Aˆn+A⊥ˆe)=ˆn×A⊥ˆe=A⊥(ˆn׈e)
The unit vector ˆn׈e lies in the plane perpendicular to ˆn and is also perpendicular to ˆe. Therefore (ˆn׈e)׈n is also a unit vector that is parallel to ˆe (by the right hand rule. So (ˆn×→A)׈n=A⊥ˆe. Thus
→A=A‖ˆn+A⊥ˆe=(→A⋅ˆn)ˆn+(ˆn×→A)׈n