Vector: Basic definition and addition

Scalar: The quantities which are specified by magnitude and no directions and follow the ordinary algebra are called scalar quantity. Example: mass, volume, speed, density etc.

Vector: The quantities which are specified by magnitude and directions and follow a special type of algebra (vector algebra) are called vector quantity. Example: velocity, acceleration, force etc.

Representation of a vector: A vector can be represented by joining two points with arrow.  represents a vector where point A is the initial point or origin and point B represents the terminal point. Arrow is directed from origin to terminal point. Let  is  and the magnitude of  is || = || = P = AB.

Classification of vector:

(i) Equal vector: If two vectors have equal magnitude and same direction then those vectors are called equal vector. Here (= ) and (= ) are of same magnitude and same direction. So, = .

(ii) Opposite vectors: If two vectors have equal magnitude but opposite direction then those vectors are called opposite vector. Here (= ) and (= – ) have equal magnitude but opposite direction then = – .

(iii) Collinear vectors: If all the vectors are parallel to each other whatever be their magnitude then those vectors are collinear vectors. Here ,  and  are parallel so they are collinear vectors.

(iv) Null vector: The vector which having magnitude zero and the direction is undefined then the vector is called null vector.

(v) Orthogonal vector: Two vectors are said to be orthogonal if they are perpendicular to each other.

(vi) Unit vector: The vector which having magnitude one and the direction is towards the given vector is called unit vector. If is a vector then the unit vector = .

If = x + y + z then the unit vector =  (where , and are the unit vectors along X, Y, Z axis respectively.)

Triangle law of vector addition:

Statement: If two vectors are represented both in magnitude and direction by two sides of a triangle taken in the same order, then the resultant of these vectors is represented both in magnitude and direction by the third side of the triangle taken in the opposite order.

Prove: Let us consider  and  acting simultaneously on a particle be represented both in magnitude and direction by two sides OA and AC of OAC taken in the same order.  is the resultant of  and  is represented by side OC.

So,  = +

CN is the perpendicular on ON and θ is the angle between  and .

In right-angled ONC, OC2 = ON2 +NC2

Or, R2 = (OA + AN)2 + NC2 

Or, R2 = OA2 + AN2 + 2OA.AN + NC2 

Or, R2 = OA2 + AN2 + 2OA.ACcos + NC2

Or, R2 = OA2 + AC2 + 2OA.ACcos

[From right-angled ANC,  =  so, AN = AC  and AN2+ NC2 = AC2]

 R2 = P2 + Q2 + 2PQ

Let us consider creates angle  with . Then from right-angled ONC,  = = =

Or, = [From right-angled ANC,  = ].

Parallelogram law of vector addition:

Statement: If two vectors are represented both in magnitude and direction by two adjacent sides of a parallelogram drawn from a point, then their resultant will be represented both in magnitude and direction by the diagonal of the parallelogram drawn from that point.

Prove: Same as triangle law.

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