`●` This product is a vector. Two important quantities in the study of rotational motion, namely, moment of a force and angular momentum, are defined as vector products.
`color{purple}bbul{"Definition of Vector Product"}`
`=>` A vector product of two vectors `a` and `b` is a vector `c` such that
(i) magnitude of `c = c = ab sin theta` where `a` and `b` are magnitudes of `a` and `b` and `theta` is the angle between the two vectors.
(ii) `c` is perpendicular to the plane containing `a` and `b`.
(iii) if we take a right handed screw with its head lying in the plane of `a` and `b` and the screw perpendicular to this plane, and if we turn the head in the direction from `a` to `b`, then the tip of the screw advances in the direction of `c`. This right handed screw rule is illustrated in Fig. 7.15a.
`=>` Alternately, if one curls up the fingers of right hand around a line perpendicular to the plane of the vectors `a` and `b` and if the fingers are curled up in the direction from `a` to `b`, then the stretched thumb points in the direction of `c`, as shown in Fig. 7.15b.
`color{green}☞` A simpler version of the right hand rule is the following : Open up your right hand palm and curl the fingers pointing from `a` to `b`. Your stretched thumb points in the direction of `c`.
`color{green}☞` It should be remembered that there are two angles between any two vectors `a` and `b` . In Fig. 7.15 (a) or (b) they correspond to `theta` (as shown) and `(360^0– theta)`. While applying either of the above rules, the rotation should be taken through the smaller angle `(< 180^0)` between `a` and `b`. It is `theta` here.
`●` Because of the cross used to denote the vector product, it is also referred to as cross product.
`►` Note that scalar product of two vectors is commutative as said earlier, `a*b = b*a`
`●` The vector product, however, is not commutative, i.e. `a xx b ne b xx a` The magnitude of both `a xx b` and `b xx a` is the same `( ab sin theta )` ; also, both of them are perpendicular to the plane of `a` and `b`. But the rotation of the right-handed screw in case of `a xx b` is from `a` to `b`, whereas in case of `b xx a` it is from `b` to `a`. This means the two vectors are in opposite directions. We have
`color{blue}{a xx b=-b xx a}`
`►` Another interesting property of a vector product is its behaviour under reflection. Under reflection (i.e. on taking the mirror image) we have `x -> x,y -> y` and `z -> z` . As a result all the components of a vector change sign and thus `a ->-a, b -> −b` . What happens to `a xx b` under reflection?
`a xx b->(−a)xx (−b) = a xx b`
Thus, `a xx b` does not change sign under reflection.
`►` Both scalar and vector products are distributive with respect to vector addition.
Thus,
`a*(b + c) = a*b + a*c`
`color{blue}{a xx (b + c) = a xx b + a xx c}`
`►` We may write `c = a xx b` in the component form. For this we first need to obtain some elementary cross products:
(i) `a xx a = 0` (`0` is a null vector, i.e. a vector with zero magnitude)
This follows since magnitude of `a xx a` is
`a^2 sin 0^o=0`
From this follow the results
`hat i xx hat i =0, hat j xx hat j=0, hat k xx hat k =0`
(ii) `hat i xx hat j = hat k`
`●` Note that the magnitude of `hat i × hat j` is `sin90^0` or `1`, since `hat i` and `hat j` both have unit magnitude and the angle between them is `90^0`. Thus, `hat i × hat j` is a unit vector. A unit vector perpendicular to the plane of `hat i` and `hat j` and related to them by the right hand screw rule is `hat k` . Hence, the above result. You may verify similarly,
`hat j xx hat k = hat i` and `hat k xx hat i = hat j`
From the rule for commutation of the cross product, it follows:
`hat j xx hat i = −hat k, hat k × hat j = −hat i, hat i × hat k = −hat j`
`●` Note if `hat i,hat j, hat k` occur cyclically in the above vector product relation, the vector product is positive. If `hat i , hat j , hat k` do not occur in cyclic order, the vector product is negative.
Now, `a xx b = (a_x hat i + a_y hat j + a_z hat k) xx( b_x hat i + b_y hat j + b_z hat k)`
` = a_x b_y hat k − a_x b_z hat j − a_y b_x hat k + a_y b_z hat i + a_z b_x hat j − a_z b_y hat i`
`color{red}{axxb=( a_y b_z − a_z b_ i)hati + (a_z b_x − a_x b_z ) hat j + (a_x b_y − a_y b_x)hat k}`
`=>` We have used the elementary cross products in obtaining the above relation. The expression for `a xx b` can be put in a determinant form which is easy to remember.
`color{orange}{a xx b = |(hat i , hat j , hat k),(a_x , a_y , a_z),(b_x, b_y , b_z)|}`
`●` This product is a vector. Two important quantities in the study of rotational motion, namely, moment of a force and angular momentum, are defined as vector products.
`color{purple}bbul{"Definition of Vector Product"}`
`=>` A vector product of two vectors `a` and `b` is a vector `c` such that
(i) magnitude of `c = c = ab sin theta` where `a` and `b` are magnitudes of `a` and `b` and `theta` is the angle between the two vectors.
(ii) `c` is perpendicular to the plane containing `a` and `b`.
(iii) if we take a right handed screw with its head lying in the plane of `a` and `b` and the screw perpendicular to this plane, and if we turn the head in the direction from `a` to `b`, then the tip of the screw advances in the direction of `c`. This right handed screw rule is illustrated in Fig. 7.15a.
`=>` Alternately, if one curls up the fingers of right hand around a line perpendicular to the plane of the vectors `a` and `b` and if the fingers are curled up in the direction from `a` to `b`, then the stretched thumb points in the direction of `c`, as shown in Fig. 7.15b.
`color{green}☞` A simpler version of the right hand rule is the following : Open up your right hand palm and curl the fingers pointing from `a` to `b`. Your stretched thumb points in the direction of `c`.
`color{green}☞` It should be remembered that there are two angles between any two vectors `a` and `b` . In Fig. 7.15 (a) or (b) they correspond to `theta` (as shown) and `(360^0– theta)`. While applying either of the above rules, the rotation should be taken through the smaller angle `(< 180^0)` between `a` and `b`. It is `theta` here.
`●` Because of the cross used to denote the vector product, it is also referred to as cross product.
`►` Note that scalar product of two vectors is commutative as said earlier, `a*b = b*a`
`●` The vector product, however, is not commutative, i.e. `a xx b ne b xx a` The magnitude of both `a xx b` and `b xx a` is the same `( ab sin theta )` ; also, both of them are perpendicular to the plane of `a` and `b`. But the rotation of the right-handed screw in case of `a xx b` is from `a` to `b`, whereas in case of `b xx a` it is from `b` to `a`. This means the two vectors are in opposite directions. We have
`color{blue}{a xx b=-b xx a}`
`►` Another interesting property of a vector product is its behaviour under reflection. Under reflection (i.e. on taking the mirror image) we have `x -> x,y -> y` and `z -> z` . As a result all the components of a vector change sign and thus `a ->-a, b -> −b` . What happens to `a xx b` under reflection?
`a xx b->(−a)xx (−b) = a xx b`
Thus, `a xx b` does not change sign under reflection.
`►` Both scalar and vector products are distributive with respect to vector addition.
Thus,
`a*(b + c) = a*b + a*c`
`color{blue}{a xx (b + c) = a xx b + a xx c}`
`►` We may write `c = a xx b` in the component form. For this we first need to obtain some elementary cross products:
(i) `a xx a = 0` (`0` is a null vector, i.e. a vector with zero magnitude)
This follows since magnitude of `a xx a` is
`a^2 sin 0^o=0`
From this follow the results
`hat i xx hat i =0, hat j xx hat j=0, hat k xx hat k =0`
(ii) `hat i xx hat j = hat k`
`●` Note that the magnitude of `hat i × hat j` is `sin90^0` or `1`, since `hat i` and `hat j` both have unit magnitude and the angle between them is `90^0`. Thus, `hat i × hat j` is a unit vector. A unit vector perpendicular to the plane of `hat i` and `hat j` and related to them by the right hand screw rule is `hat k` . Hence, the above result. You may verify similarly,
`hat j xx hat k = hat i` and `hat k xx hat i = hat j`
From the rule for commutation of the cross product, it follows:
`hat j xx hat i = −hat k, hat k × hat j = −hat i, hat i × hat k = −hat j`
`●` Note if `hat i,hat j, hat k` occur cyclically in the above vector product relation, the vector product is positive. If `hat i , hat j , hat k` do not occur in cyclic order, the vector product is negative.
Now, `a xx b = (a_x hat i + a_y hat j + a_z hat k) xx( b_x hat i + b_y hat j + b_z hat k)`
` = a_x b_y hat k − a_x b_z hat j − a_y b_x hat k + a_y b_z hat i + a_z b_x hat j − a_z b_y hat i`
`color{red}{axxb=( a_y b_z − a_z b_ i)hati + (a_z b_x − a_x b_z ) hat j + (a_x b_y − a_y b_x)hat k}`
`=>` We have used the elementary cross products in obtaining the above relation. The expression for `a xx b` can be put in a determinant form which is easy to remember.
`color{orange}{a xx b = |(hat i , hat j , hat k),(a_x , a_y , a_z),(b_x, b_y , b_z)|}`