Dot and Cross Product of Vector

Scalar product or dot product:

The scalar product of two vectors and  is defined as the product of the magnitudes of and  and the cosine of the angle between them.

If A and B creates angle θ then, . = AB .

Properties of dot product:

(i) . = BA  = AB  = .

(ii) If the angle between and  is  then, .   = ABcos  = AB

(iii) . = AAcos =

(iv) If the angle between and is then, . = ABcos  = 0

(v) . = . = . = 1.1cos = 1

(vi) . = . = . = 1.1cos = 0

Concepts on dot product:

1. WORK DONE:

A force  = (2 + 3) N is applied on a body to move it from point P (1, 2, -1) m to point Q (2, 4, 1) m. What is the work done required?

Force  = (2 + 3)N and displacement of the body = = (2 – 1 ) + (4 – 2) + (1 + 1) = (+ 2 + 2) m.

Work done W = . = (2 + 3).( + 2 + 2) = 2 + 6 – 2 = 6J.

2. ANGLE BETWEEN TWO VECTORS:

If  is the angle between and  then  = .

EXAMPLE: What is the angle between = 2+ 3 + and = + 2 – 2?

Here =  =  = .

Vector product or cross product: The vector product of two vectors and is defined as the product of the magnitudes of and and the sine of the angle between them.

If and creates angle θ then = AB. Where is the unit vector perpendicular to the plane of and .

Properties of cross product:

 = BA (-) and ||= BA = AB = ||

(ii) If the angle between and is  then = ABsin = 0

(iii) = A.Asin = 0

(iv) If the angle between and  is then = ABsin = AB.

(v) = = = 1.1sin = 0

(vi) = , = = .

CONCEPTS ON CROSS PRODUCT:

1. AREA:

Area of a triangle: and are represented by two sides AB and AC of ABC and CD is perpendicular to AB. Angle between and is .

The area of ABC = CD.AB = ACsin.AB = QsinP = PQsin = .

The direction of area vector is perpendicular outward to the plane of the triangle.

2. TORQUE: Torque or moment of force about point Q = Force at point P Perpendicular distance QO

= F.QPsin = FRsin = RFsin.

=

The direction of torque is perpendicular outward to the plane of and .

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