Vector quantity and its representation:Vector quantity describes the motion of objects in both magnitude and direction. Some of such quantities are Displacement, Velocity, Acceleration, and Momentum. It is often represented using ‘Scaled Vector diagrams’ where size of the vector arrow represents its magnitude and vector arrow the direction.
Head to Tail method to add and subtract vectors:It is worth mentioning here that vectors can only be added geometrically or graphically and they can’t be added using ordinary Algebra since they have both magnitude and direction. To discuss vector addition and subtraction using this method, let us consider two vectors A and B acting along the same line. To add these two vectors, the tail of Vector B is joined with the head of Vector A.
The sum of two vectors is the Resultant vector R which acts in the same direction as that of vectors A and B. Subtraction of any two vectors is similar to their addition. But the Resultant vector acts in the same direction as that of the vector with greater magnitude. Vector Multiplication:Scalar product and Vector product are two different kinds of Vector multiplication. The scalar product of two vectors A and B is equal to the product of their magnitudes and the cosine of the angle between them. Hence it is called as the dot product of vector. If the two vectors are parallel to each other, then the angle between them is zero.
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Thus the dot product of parallel vectors becomes equal to the product of their magnitudes. The vector product of two vectors C and D is equal to the product of their magnitudes and the sine of the smaller angle between them. The direction of the resultant vector is perpendicular to a plane containing the two vectors. Hence it is called as the vector cross product and the vector cross product formula can be given as follows:(Vector C) X (Vector D) equals to (Magnitude of Vector C) X (Magnitude of Vector D) X (Sine of angle between Vectors C and D) X (Unit Vector).A new term ‘Unit Vector’, mentioned in the above formula, is a vector perpendicular to the plane formed by two vectors.
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The cases of unit vector cross product are discussed below: The cross product of any unit vector with itself is zero since the angle between them is zero and the cross product of any different two unit vectors is one since the angle between them is 90 degrees.