Subtraction of vectors:
Let us consider and creates angle θ. To subtract from we can write + (- ) [where – is the opposite vector of ]. The angle between and – is (180 – θ) and is the resultant of and – .
The magnitudes of is R2 = P2 + Q2+2P.Q
Or, R2 = P2 +Q2– 2PQ
If creates angle with then tan =
Or, tan = .
Resolution of vectors:
Resolution of a vector into its components is the process to determining a set of vectors, whose resultant is the given vector. Each vector in that set is called a component of the given vector.
Prove: Let us consider (= ) is resolute along the line OA and OB creates angles and with OC respectively. From point C, CD and CE are drawn parallel to OB and OA respectively. OD and OE represent and respectively.
From ODC, COD = , OCD = , then ODC = 1800– ( + ).
From trigonometry, we can write = =
Or, = =
Or, = =
So, P = and Q = .
Resolution of a vector into two rectangular components:
Let us consider OX and OY are two mutually perpendicular axes, where O is the origin. OB represents creates angle θ with +ve x axis. Let BA and BC are the perpendiculars on OX and OY axis respectively. and are the components of and represented by OA and OC respectively. I.e. = + .
The magnitudes of and are represented as P = R [ = ]
and Q = R [ = ]
So that R = and tan = .
Representation of a vector by coordinates:
Let us consider OX, OY and OZ are three perpendicular axes, where O is the origin. Let P is a point with coordinates (x, y, z) and = is the position vector of P. , and are the unit vectors along + ve X, Y and Z axis. Let PB is the perpendicular drawn on x-y plane. BA and BC are the perpendicular drawn on X and Y axis respectively. PD is the perpendicular drawn on Z axis.
Therefore = x, = = y and = = z and their magnitudes are OA = x, AB = OC = y and BP = OD = z.
From OAB, = + and from OBP, = +
Or, = + +
So, = x + y + z.
Magnitude: From OAB, OB2 = OA2 + AB2 and from OBP, OP2 = OB2 + BP2
Or, OP2 = OA2 + AB2 + BP2
So, R2 = x2 + y2 + z2
Direction: Let us consider , and are the angles of R with +ve X, Y and Z axis respectively. Then, cos = , cos = , cos = where cos, cos and cos are the direction cosines of .
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