But since E, G, D, F are in one plane, and

2m (1 − n) OE − (1 − n) OG + 2mnOD—mOF = 0,

we must have (34: 8)

COR. The plane cuts other two edges at F and G, so that

ADDITIONAL EXAMPLES TO CHAP. IV.

1. Straight lines are drawn terminated by two given straight lines, to find the locus of a point in them whose distances from the extremities have a given ratio.

2. Two lines and a point S are given, not in one plane; find the locus of a point P such that a perpendicular from it on one of the given lines intersects the other, and the portion of the perpendicular between the point of section and P bears to SP a constant ratio. Prove that the locus of P is a surface of the second order.

3. Prove that the section of this surface by a plane perpendicular to the line to which the generating lines are drawn perpendicular is a circle.

4. Prove that the locus of a point whose distances from two given straight lines have a constant ratio is a surface of the second order.

5. A straight line moves parallel to a fixed plane and is terminated by two given straight lines not in one plane; find the locus of the point which divides the line into parts which have a constant ratio,

6. Required the locus of a point P such that the sum of the projections of OP on OA and OB is constant.

7. If the sum of the perpendiculars on two given planes from the point A is the same as the sum of the perpendiculars from B, this sum is the same for every point in the line AB.

8. If the sum of the perpendiculars on two given planes from each of three points A, B, C (not in the same straight line) be the same, this sum will remain the same for every point in the plane ABC.

9. A solid angle is contained by four plane angles. Through a given point in one of the edges to draw a plane so that the section shall be a parallelogram.

10. Through each of the edges of a tetrahedron a plane is drawn perpendicular to the opposite face. Prove that these planes pass through the same straight line.

11. ABC is a triangle formed by joining points in the rectangular co-ordinates OA, OB, OC; OD is perpendicular to ABC. Prove that the triangle AOB is a mean proportional between the triangles ABC, ABD.

12. Vap VBp+ (Vaß)=0 is the equation of a hyperbola in p, the asymptotes being parallel to a, ß.

These are the three forms of the vector equation.

Form (2) may be written

p2 - 2.Sap=0.

If OC=c, form (3) may be written

p2 - 2Syp=c2 - a3.

EXAMPLES.

37. Ex. 1.

Ex. 1. The angle in a semicircle is a right angle.

.(3).

therefore p, p-2a are vectors at right angles to one another. But p-2a is DP ;

... DPA is a right angle.

Ex. 2. If through any point O within or without a circle, a straight line be drawn cutting the circle in the points P, Q, the product OP. OQ is always the same for that point.

The third form of the equation may be written

(Tp)2 +2TpSyUp + c2 − a2 = 0,

which shews that Tp has two values corresponding to each value of Up, the product of which is c2-a. Therefore, &c.

Ex. 3. If two circles cut one another, the straight line which joins the points of section is perpendicular to the straight line which joins the centres.

Let O, C be the centres, P, Q the points of section ;

Ex. 4. O is a fixed point, AB a given straight line. A point Q is taken in the line OP drawn to a point P in AB, such that

to find the locus of Q.

then

or

OP. OQ = k*;

Let OA perpendicular to AB be a, vector a ;