Vector (cross) product *

Transcription

1 OpenStax-CNX module: m Vector (cross) product * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Abstract Vector multiplication provides concise and accurate representation of natural laws, which involve vectors. The cross product of two vectors a and b is a third vector. The magnitude of the vector product is given by the following expression : c = a b = absinθ (1) where θ is the smaller of the angles between the two vectors. It is important to note that vectors have two angles θ and 2π - θ. We should use the smaller of the angles as sine of θ and 2π - θ are dierent. If "n" denotes unit vector in the direction of vector product, then c = a b = absinθ n (2) 1 Direction of vector product The two vectors (a and b) dene an unique plane. The vector product is perpendicular to this plane dened by the vectors as shown in the gure below. The most important aspect of the direction of cross product is that it is independent of the angle, θ, enclosed by the vectors. The enclosed angle, θ, only impacts the magnitude of the cross product and not its direction. * Version 1.10: Feb 12, :46 pm

2 OpenStax-CNX module: m Direction of vector product Figure 2 Incidentally, the requirement for determining direction suits extremely well with rectangular coordinate system. We know that rectangular coordinate system comprises of three planes, which are at right angles to each other. It is, therefore, easier if we orient our coordinate system in such a manner that vectors lie in one of the three planes dened by the rectangular coordinate system. The cross product is, then, oriented in the direction of axis, perpendicular to the plane of vectors.

3 OpenStax-CNX module: m Direction of vector product Figure 2 As a matter of fact, the direction of vector product is not yet actually determined. We can draw the vector product perpendicular to the plane on either of the two sides. For example, the product can be drawn either along the positive direction of y axis or along the negative direction of y-axis (See Figure below).

4 OpenStax-CNX module: m Direction of vector product Figure 2 The direction of the vector product, including which side of the plane, is determined by right hand rule for vector products. According to this rule, we place right st such that the curl of the st follows as we proceed from the rst vector, a, to the second vector, b. The stretched thumb, then gives the direction of vector product.

5 OpenStax-CNX module: m Direction of vector product Figure 2 When we apply this rule to the case discussed earlier, we nd that the vector product is in the positive y direction as shown below :

6 OpenStax-CNX module: m Direction of vector product Figure 2 Here, we notice that we move in the anti-clockwise direction as we move from vector, a, to vector, b, while looking at the plane formed by the vectors. This fact can also be used to determine the direction of the vector product. If the direction of movement is anticlockwise, then the vector product is directed towards us; otherwise the vector product is directed away on the other side of the plane. It is important to note that the direction of cross product can be on a particular side of the plane, depending upon whether we take the product from a to b or from b to a. This implies : a b b a Thus, vector product is not commutative like vector addition. It can be inferred from the discussion of direction that change of place of vectors in the sequence of cross product actually changes direction of the product such that : a b = b a (3) 2 Values of cross product The value of vector product is maximum for the maximum value of sinθ. Now, the maximum value of sine is sin 90 = 1. For this value, the vector product evaluates to the product of the magnitude of two vectors

7 OpenStax-CNX module: m multiplied. Thus maximum value of cross product is : ( a b ) max = ab (4) The vector product evaluates to zero for θ = 0 and 180 as sine of these angles are zero. These results have important implication for unit vectors. The cross product of same unit vector evaluates to 0. i i = j j = k k = 0 (5) The cross products of combination of dierent unit vectors evaluate as : i j = k ; j k = i ; k i = j j i = k ; k j = i ; i k = j (6) There is a simple rule to determine the sign of the cross product. We write the unit vectors in sequence i,j,k. Now, we can form pair of vectors as we move from left to right like i x j, j x k and right to left at the end like k x i in cyclic manner. The cross products of these pairs result in the remaining unit vector with positive sign. Cross products of other pairs result in the remaining unit vector with negative sign. 3 Cross product in component form Two vectors in component forms are written as : a = a x i + a y j + a z k b = b x i + b y j + b z k In evaluating the product, we make use of the fact that multiplication of the same unit vectors gives the value of 0, while multiplication of two dierent unit vectors result in remaining vector with appropriate sign. Finally, the vector product evaluates to vector terms : a b = ( a x i + a y j + a z k ) ( b x i + b y j + b z k ) a b = a x i b y j + a x i b z k + a y j b x i + a y j b z k + a z k b x i + a z k b y j a b = a x b x k a x b z j a y b x k + a y b z i + a z b x j a z b y i a b = ( a y b z a z b y ) i + ( a z b x a x b z ) j + ( a x b y a y b x ) k (7) Evidently, it is dicult to remember above expression. If we know to expand determinant, then we can write above expression in determinant form, which is easy to remember. a b = i j k a x a y a z (8) b x b y b z Exercise 1 (Solution on p. 11.) If a = 2i + 3j and b = -3i 2j, nd A x B. Exercise 2 (Solution on p. 11.) Consider the magnetic force given as : F = q (v x B) Given q = 10 6 C, v = (3i + 4j) m/s, B = 1i Tesla. Find the magnetic force.

8 OpenStax-CNX module: m Geometric meaning vector product In order to interpret the geometric meaning of the cross product, let us draw two vectors by the sides of a parallelogram as shown in the gure. Now, the magnitude of cross product is given by : Cross product of two vectors Figure 8: Two vectors are represented by two sides of a parallelogram. a b = absinθ We drop a perpendicular BD from B on the base line OA as shown in the gure. From OAB, Substituting, we have : bsinθ = OBsinθ = BD a b = OA x BD = Base x Height = Area of parallelogram It means that the magnitude of cross product is equal to the area of parallelogram formed by the two vectors. Thus, Area of parallelogram = a b (9) Since area of the triangle OAB is half of the area of the parallelogram, the area of the triangle formed by two vectors is : Area of triangle = 1 2 x a b (10)

9 OpenStax-CNX module: m Attributes of vector (cross) product In this section, we summarize the properties of cross product : 1: Vector (cross) product is not commutative a b b a A change of sequence of vectors results in the change of direction of the product (vector) : a b = b a The inequality resulting from change in the order of sequence, denotes anti-commutative nature of vector product as against scalar product, which is commutative. Further, we can extend the sequence to more than two vectors in the case of cross product. This means that vector expressions like a x b x c is valid. Ofcourse, the order of vectors in sequence will impact the ultimate product. 2: Distributive property of cross product : a ( b + c ) = a b + a c 3: The magnitude of cross product of two vectors can be obtained in either of the following manner : or, or, a b = a x ( bsinθ ) a b = absinθ a b = a x component of b in the direction perpendicular to vector a a b = b x ( asinθ ) b a = a x component of a in the direction perpendicular to vector b 4: Vector product in component form is : a b = i j k a x a y a z b x b y b z 5: Unit vector in the direction of cross product Let n be the unit vector in the direction of cross product. Then, cross product of two vectors is given by : a b = absinθ n a b = a b n n = a b a b 6: The condition of two parallel vectors in terms of cross product is given by :

10 OpenStax-CNX module: m a b = absinθ n = absin0 n = 0 If the vectors involved are expressed in component form, then we can write the above condition as : a b = i j k a x a y a z = 0 b x b y b z Equivalently, this condition can be also said in terms of the ratio of components of two vectors in mutually perpendicular directions : a x bx = ay b y = az b z 7: Properties of cross product with respect to unit vectors along the axes of rectangular coordinate system are : i i = j j = k k = 0 i j = k ; j k = i ; k i = j j i = k ; k j = i ; i k = j

11 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 7) a b = ( 2i + 3j ) ( 3i 2j ) Neglecting terms involving same unit vectors, we expand the multiplication algebraically as : a b = ( 2i ) ( 2j ) + ( 3j ) ( 3i ) a b = 4k + 9k = 5k Solution to Exercise (p. 7) F = q ( v B ) = 10 6 { ( 3i + 4j ) ( 1i ) } F = 4 x 10 6 k