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Vector Analysis 1

vectors, ab, parallel, cd, quantity and scalar

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VECTOR ANALYSIS. 1. A quantity which is related to a definite direction in space is called a vector quantity, e.g., the displacement of a point, velocity, force, axis of rotation, elec tric current; a quantity having no reference to direction is called a scalar quantity, e.g., mass, volume, time, energy, electric charge. A vector quantity can be represented by a line segment if (1) the length of the segment is taken to rep resent on an assigned scale the magnitude of the quantity; (2) the segment be placed in die proper direction; and (3) be given an arrow head showing in which of the two opposite senses it is regarded as pointing. Such a di rected line segment is called a vector. In ac cordance with etymology the vector AB may be recorded as the operator which carries a moving point from A to B, and is represented by an arrow reaching from its initial to its terminal point. (Bold face type is used for vectors).

2. Equal The vectors AB and CD are said to be equal when the segments AB and CD are equal and parallel, and the arrow heads from A to B and from C to D point in the same sense. If AB and CD are equal and parallel, but the arrow heads are in opposite senses we say that 3. Addition of Vectors. Two vectors AB and CD are added by drawing through the ter minal point of the first another vector BE equal to the secon I; the vector AE is then called the sum of AB and BE, and also of AB and CD. The annexed parallelogram shows that vector addition is commutative, that is a + b=b + a. Similarly for the addition 0A=a, drawn from a fixed origin, and A may be called the point whose vector is a, or the point a. The indefinite line AB is denoted by (a, b). The vector AB equals ba, that is, terminal vector minus initial vector; similarly BA equals ab.

7. Vector Equation of a Line. Let OP=r be the vector of a running point on the line (a, b) ; then since OP=OA + AP and AP=x AB, where x is the scalar expressing the ratio of AP to AB, hence r=a x (ba). This equation gives the vector of P in terms of a varying scalar x.

of three or more vectors. In any closed polygon ABCDE we have AB BC + CD + DE=-AE and AB + BC + CD + DE + EA=AA= 0 whether the vertices are in one plane or not The difference of two vectors is the sum of the first and the opposite of the second, that is ab=a + (b).

4. Resolution of Vectors. Vectors are called coplanar when they are parallel to the same plane. Any vector r, coplanar with two non-parallel vectors a and b, can be resolved into two components parallel to a and b, by constructing a parallelogram whose diagonal is r and whose sides are parallel to a and b; hence r can be expressed in the form r=xa + yb, where x and y are scalar numbers express ing the ratios of the two component vectors to the parallel vectors a and b. Similarly by con structing a parallelopiped, r may be resolved into three components parallel tc three given non-coplanar vectors a, 13, c, in the form r=xa + yb sc.

It is usual to take instead of a, b, c, three vectors 1, j, k of unit length along three rect angular axes OX, OY, OZ, then r=xi+yj-l-sk, where x, y, z are the co-ordinates of the terminal point of r, its initial point being at O. The axes are usually taken to form a right handed system, positive rotation about OX, OY and OZ respectively carrying OY to OZ, OZ to OX, and OX to OY.

5. Conditions for Equality.A vector equation r=e, that is, xa yb + sc=x'a + y'b dc, where a, b, c are non-coplanar, neces sitates the three scalar equations x=x', 0=,ff'. In two dimensions, the equation xa + yt=x'a y'b, where a and b are not parallel, necessitates the two scalar equations x=x', Y=Y' 6. Vector of a Point. The position of a point A may be defined by means of its vector Example 1.If the segment b) is di vided as to tn2, the vector of P (tn, b + m2a)/mi Example If masses m,, in: are placed at their centre of gravity is (ntiai mai)/mi+ tn2. Extend this to three masses nil tni, nis at al, a2, as.

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