Tamil Nadu Board of Secondary EducationHSC Science Class 11th

Discuss the properties of scalar and vector products. - Physics

Discuss the properties of scalar and vector products.

Solution

Properties of the scalar product of two vectors are:

(1) The product quantity vecA . vecB is always a scalar. It is positive if the angle between the vectors is acute (i.e., < 90°) and negative if the angle between them is obtuse (i.e. 90°< 0 < 180°).

(2) The scalar product is commutative, i.e. vecA vecB  ± vecB . vecA

(3) The vectors obey distributive law i.e. vecA(vecB + vecC) = vecA . vecB + vecA . vecC

(4) The angle between the vectors θ = cos^-1[(vecA.vecB)/(AB)]

(5) The scalar product of two vectors will be maximum when cos θ = 1, i.e. θ = 0°, i.e., when the vectors are parallel;

(vecA . vecB)_"max" = AB

(6) The scalar product of two vectors will be minimum, when cos θ = -1, i.e. θ = 180°.

(vecA . vecB)_"min" = -AB when the vectors are anti-parallel.

(7) If two vectors vecA and vecB are perpendicular to each other then their scalar product vecA . vecB = 0, because cos 90° 0. Then the vectors vecA and vecB are said to be mutually orthogonal.

(8) The scalar product of a vector with itself is termed as self-dot product and is given by (vecA)^2 = vecA . vecA = A A cos 0 = A^2. Here angle 0 = 0°.

The magnitude or norm of vector vecA is |vecA| = A = sqrt(vecA . vecA.)

(9) In case of a unit vector hatn
hatn . hatn = 1 × 1 × cos 0 = 1. For example, hati - hatj = hatj . hatj = hatk . hatk = 1.

(10) In the case of orthogonal unit vectors, hati, hatj and hatk, hati . hatj = hatj . hatk = hatk . hat i = 1.1 cos 90° = 0

(11) In terms of components the scalar product of vecA and vecB can be written as vecA . vecB = (A_xhati + A_yhatj + A_zhatk) . (B_xhati + B_yhatj + B_Zhatk)

= AxBx + AyBy+ AzBz, with all other terms zero.

The magnitude of vector |vecA| is given by

|vecA| = A = sqrt(A_x^2 + A_y^2 + A_z^2)

Properties of the vector product of two vectors are:

(1) The vector product of any two vectors is always another vector whose direction is perpendicular to the plane containing these two vectors, i.e., orthogonal to both the vectors vecA and vecB, even though the vectors vecA and vecB may or may not be mutually orthogonal.

(2) The vector product of two vectors is not commutative, i.e., vecA xx vecB ≠ vecB xx vecA. But,vecA xx vecB = -vecB xx vecA.

Here it is worthwhile to note that |vecA xx vecB| = |vecB xx vecA| = AB sin 0 i.e., in the case of the product vectors vecB = -vecB  "and"  vecB xx vecA, the magnitudes are equal but directions are opposite to each other.

(3) The vector product of two vectors will have maximum magnitude when sin 0 = 1, i.e., 0 = 90° i.e. when the vectors vecA and vecB are orthogonal to each other.

(vecA xx vecB)_"maz" = AB  hatn = AB  hatn

(4) The vector product of two non-zero vectors will be minimum when sin θ = 0, i.e θ = 0° or 180° (vecA xx vecB)_"min" = 0

i. e., the vector product of two non – zero vectors vanishes, if the vectors are either parallel or anti-parallel.

(5) The self–cross product, i.e., a product of a vector with itself is the null vector vecA xx vecA = AA sin 0° hatn = vec0 In physics, the null vector 0 is simply denoted as zero.

(6) The self–vector products of unit vectors are thus zero. hati xx hati = hatj xx hatj = hatk xx hatk = 0

(7) In the case of orthogonal unit vectors, hati, hatj, hatk, in accordance with the right-hand screw rule:
hati xx hatj = hatk, hatj xx hatk = hati  "and"  hatk xx hati = hatj

Also, since the cross product is not commutative, hatj xx hati = -hatk. hatk xx hatj = -hati and hati xx hatk = hatj

(8) In terms of components, the vector product of two vectors vecA and vecB is -

vecA xx vecB = |(hati     hatj     hat k), (A_x  A_y  A_z), (B_x  B_y  B_z)|

= hati(A_yB_z - A_zB_y) + hatj(A_zB_x - A_xB_z) + hatk(A_xB_y - A_yB_x)

Note that in the hatj^"th" component the order of multiplication is different than hati^"th" and hatk^"th" components.

(9) If two vectors vecA and vecB form adjacent sides in a parallelogram, then the magnitude of |vecA xx vecB| will give the area of the parallelogram as represented graphically in figure.

|vecA xx vecB| = |vecA| |vecB| sintheta

Area of parallelogram

(10) Since we can divide a parallelogram into two equal triangles as shown in the figure, the area of a triangle with vecA and vecB as sides is 1/2 |vecA xx vecB|. This is shown in the Figure. A number of quantities used in Physics are defined through vector products. Particularly physical quantities representing rotational effects like torque, angular momentum, are defined through vector products.

Area of triangle

Concept: Elementary Concept of Vector Algebra
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