Dot Product of Two Vectors

Aug 08, 2022, 16:45 IST

SCALAR (OR DOT) PRODUCT OF TWO VECTORS

Vectors & Scalars Quantities

Scalar: A scalar quantity is defined as a quantity that has magnitude only. Typical Illustrations of scalar quantities are time, speed, temperature, and volume. A scalar quantity or parameter has no directional component, only magnitude. For Illustration, the units for time (minutes, days, hours, etc.) represent an amount of time only and tell nothing of direction. Additional Illustrations of scalar quantities are density, mass, and energy.

Vector: A physical quantity which has magnitude and specific direction and which follows vector law of addition is called vector. For Illustration displacement, force, acceleration, electric field & torque etc.

Graphical Representation Of Vectors

To work with vector quantities, one must know the method for representing these quantities. Magnitude, or “size” of a vector, is also referred to as the vector’s “displacement.” It can be thought of as the scalar portion of the vector and is represented by the length of the vector. By definition, a vector has both magnitude and direction. Direction indicates how the vector is oriented relative to some reference axis, as shown in Figure.

Using north/south and east/west reference axes, vector “A” is oriented in the NE quadrant with a direction of 45 north of the o EW axis. G iving direction to scalar “A” makes it a vector. The length of “A” is representative of its magnitude or displacement.

Different Kinds Of Vectors

Parallel vectors: If two vectors have the same direction called parallel vectors. In figure are parallel vectors.

Anti Parallel Vectors: Two vectors have opposite direction called anti parallel vectors. In figure are anti parallel vectors.

Equal vectors: “If two or more vectors have equal magnitude and acting in the same direction, they are said to be equal vectors”. The two vectors shown in figure have equal length and same orientation.

Hence they represent two equal vectors even though they have at different initial points terminal points.

Negative vector: If two vectors are such that they have equal magnitude but opposite directions, each vector is negative of the other.Thus or .

Null vector: “A vector of zero magnitude is called zero vector or null vector”. It is represented by . The initial point and terminal point of the null vector coincide. Its direction is indeterminate.

Unit Vector: “A vector of unit magnitude is called unit vector”. The unit vector in the direction of given vector is obtained by dividing the given vector with its magnitude. It is conventional to denote unit vector with a “cap” instead of “bar” over the symbol. Thus if is a given vector, the unit vector in the direction of is written as (where is read as A cap or A hat)

Note: In the right handed Cartesian coordinate system, are choosen as unit vectors along, the X-axis, Y-axis and Z-axis respectively.

Co-planar Vector:Vectors, which are in the same plane are called co-planar vectors.

Non Co-planar Vector:Vectors, which are in different planes are called non-co-planar vectors.

Position Vector:“The vector used to specify the position of a point with respect to some fixed point (say origin ‘O’) is called position vector”. It is denoted as .

Consider a point ‘A’ with coordinates x, y, z in the Cartesian coordinate system. Thus the position of ‘A’ can be expressed in the vector form as . Here i, j and k are unit vectors along the X, Y and Z axes respectively. The distance of ‘A’ from the origin eventually becomes the magnitude of .

Displacement:Displacement is a vector quantity have magnitude equal to shortest distance between two points and direction from initial point to final point.

Displacement Vector:The position of the point Q with reference to the origin is represented by the position vector . Let the coordinates of the point Q are (x2, y2). Similarly represented by a position vector , let the coordinates of the point P

are (x1, y1).