a and b move on the fixed lines YOY' and X'OX respectively. The angle YOX is a right angle. The dimensions given are in inches. Draw the locus of the point

P In case I (Fig. 158) the locus is an ellipse. In case II (Fig. 159) the locus is an hyperbola.

G

CHAPTER VII

VECTOR GEOMETRY

83. Scalars, Vectors and Rotors or Locors.-A quantity which may be completely specified by stating the kind of quantity and its magnitude is called a scalar quantity or scalar. Thus the weight of a body is a scalar quantity. The weight of a body may be, say, 2 tons or 4480 pounds. The area of a plane figure is a scalar quantity. An area may be, say, 10 square feet or 1440 square inches.

In general the kind of quantity referred to is known from the name of the unit used in specifying its magnitude. Thus if a scalar quantity is 10 square feet it is known that the kind of quantity is area.

Time, temperature, volume, and energy are a few other examples of scalar quantities.

A scalar quantity is not associated with any definite direction in space but has magnitude only.

A scalar quantity may be represented by the length of a straight line drawn to scale, such line being drawn anywhere and in any direction. Thus an area of 30 square feet may be represented by a line 3 inches long. In this case the scale would be 1 inch to 10 square feet.

A quantity which, while having magnitude like a scalar quantity, is also associated with a definite direction is called a vector quantity or vector. Displacement, velocity, acceleration, and force are vector quantities because they have magnitude and direction. A displacement is referred to when it is stated that a body is moved 10 miles in the direction from west to east. A velocity is referred to when it is stated that a body is moving at the rate of 10 miles an hour in the direction from north to south.

All vector quantities have magnitude and direction but they may have other qualities which distinguish them from one another and it will presently be seen that there are important propositions relating to vectors apart from qualities other than magnitude and direction which they may possess.

A vector may be represented by a straight line AB drawn to scale in a definite direction. The length AB of the line represents the magnitude of the vector and the direction of the line represents the direction of the vector, provided it is made clear whether the direction is from A to B or from B to A. A line evidently has two directions, one being exactly opposite to the other and the distinction between

them is the sense of the direction. The sense of the direction of a vector which is represented by a line is best shown by an arrow head placed on the line. If the line which represents a vector is lettered at its extremities, it might be agreed to place the letters so that in reading them in the order in which they occur in the alphabet the sense of the direction would be given. For example if the letters at the extremities of the line are A and B then the sense of the direction would be from A to B.

A quantity which in addition to having magnitude and direction also has position is called a rotor or locor. A rotor or locor is therefore a localized vector. A locor has the qualities of a vector in addition to its quality of having position. A force acting at a definite point of a body is an example of a locor. A displacement of a body from one definite position to another is another example.

It has been seen that the vector part of a locor may be represented by a straight line and in order that this straight line may also represent the locor all that is necessary is to place the straight line in the proper position.

Absolute position in space cannot be defined and the position of a point or line can only be fixed in relation to other points or lines. In considering problems on locors it is only necessary to know their relative positions.

Since vectors have direction but not position, lines which represent vectors may be placed anywhere, provided that they have the proper directions.

The following is a convenient way of specifying a vector. Let OX (Fig. 160) be a fixed direction of reference, say from west to east, and let OA be a vector whose sense is from O to A and whose magnitude OA is equal to a. Also let the angle XOA, measured in the anti-clockwise direction, be denoted by 0. Then if the vector OA is referred to as the vector A it may be specified by the equation A = a. An extension of this method to locors is described in Art. 91, p. 91.

X

FIG. 160.

84. Addition of Parallel Vectors.-Let A, B, C, and D (Fig. 161) be parallel vectors, it is required to find a single vector which is the sum or resultant of A, B, C, and D.

Draw a straight line X,X parallel to A, B, C, and D. Take a

point O in X,X. Mark off on XX a distance Oa equal to A. Observe that as the sense of A is from left to right Oa is measured to the right of O. Make ab equal to B, measuring to the right of a

because the sense of B is from left to right. Make be equal to C, measuring to the left of b because the sense of C is from right to left. Make cd equal to D, measuring to the right of c because the sense of D is from left to right. Then Od is a vector which is the sum or

=

resultant of the vectors A, B, C, and D. If R is the resultant of the vectors A, B, C, and D, then R · A+B+C+D. If a vector whose sense is from left to right is said to be positive, then a vector whose sense is from right to left would be said to be negative. Then the result of the above example would be written RA + B − C + D. Opposite signs represent opposite

senses.

Subtraction of parallel vectors is converted into addition by changing the signs of the vectors to be subtracted and then proceeding as in addition.

It should be observed that in the addition of vectors the order in which they are taken does not affect the result. For A+B+C+D A+B+D+C = B+ D + A + C.

=

85. Addition of Inclined Vectors. The Vector Polygon.Let A, B, and C (Fig. 162) be three given vectors, it is required to find a single vector which is the sum or resultant of the vectors A, B, and C.

b

CH

FIG. 162.

m

Take a point o and draw the vector oa parallel and equal to A. The sense of oa is from o to a the same as that of A. The vector A is now represented by oa. From a draw the vector ab parallel and equal to B. The sense of ab is from a to b the same as that of B. The vector B is now represented by ab. From b draw the vector be parallel and equal to C. The sense of be is from b to c the same as that of C. The vector C is now represented by bc. Join oc, then the vector oc whose sense is from o to c is the sum or resultant of the The polygon oabc is called a vector polygon. The vector polygon is of very great importance in vector geometry and it has numerous applications in mechanics.

vectors A, B, and C.

It will be useful to consider the vector polygon as applied (1) to displacements, (2) to velocities, (3) to accelerations, and (4) to forces.

(1) Suppose a man to be standing on the deck of a ship which is sailing through water which is itself moving in relation to the earth. Let the man walk across the deck along a straight line whose length and direction in relation to the ship are given by the vector A (Fig. 162). Let the vector B represent the direction and distance which the ship sails through the water in the time that the man walks the distance A. Lastly let the vector C represent the direction and distance which the water moves in the same time.

Let the vector polygon oabe (Fig. 162) be drawn. While the man is walking along oa that line is carried parallel to itself by the ship into the position mb. Consequently, neglecting for the moment the motion of the water, the man travels along the imaginary line ob over the water. But while this is happening the imaginary line ob is travelling parallel to itself with the water and reaches the position ne when the man has finished his walk. The actual displacement of the

man in relation to the earth is therefore represented by the line oc. That is, oc is the sum or resultant of the three displacements A, B, and C.

(2) Since velocity is displacement or distance moved in unit time. it follows that if t is the time taken by the man in walking the distance A, t will also be the time of the displacement B of the ship and of the displacement C of the water, and if each displacement is divided by the results are the several velocities. Hence the velocities are

oa ab be

oc

and and the polygon oabc, measured with

t

t' t' t' a suitable scale, will be a polygon of velocities in which oa represents the velocity of the man's walking on the ship, ab represents the velocity of the ship through the water, be represents the velocity of the water over the earth, and oc is the resultant velocity of the man or his velocity in relation to the earth.

(3) Let f denote the uniform acceleration or uniform rate of increase of the velocity of a moving body, then in time t the increase in velocity is ft. If A, B, and C (Fig. 162) represent three accelerations simultaneously impressed on a body these will also represent three corresponding increases in velocity and the resultant increase in velocity will be represented by oc, and oc will therefore represent the resultant acceleration. oabc is therefore an acceleration polygon.

(4) A force acting on a body causes it to move with a uniformly increasing velocity or acceleration and the magnitude of the acceleration is proportional to the magnitude of the force and takes place in the same direction as that of the force. Hence if oabc (Fig. 162) is an acceleration polygon it is also a force polygon.

If R is the sum or resultant of the vectors A, B, and C, then R = A + B + C and the order in which the vectors are taken in performing the summation is immaterial. That is, if R = A + B + C, then, also, R= B+ A+ C = C + A + B. The addition is performed by drawing the vector polygon, three sides of which are A, B, and C, and the fourth or closing side is R.

R =

0,

Observe that if R = A+B+C, then A+B+C which shows that if the sense of R is reversed the sum of the four vectors A, B, C, and R is zero.

another vector.

When the sum of a number of vectors is zero the vector polygon closes without the use of

86. Subtraction Vectors. At (7) Fig. 163, three vectors A, B, and C are given, B and C being equal and parallel but the sense of C is opposite to that of B. Hence C = −B.

If R = A - B then R = A + C.

The solution of R = A + C is shown at (m) and the solution of

R = AB is shown at (n).