Let the equation to the given plane and straight line be z = Ax+ By + C z = az + (2) where it must be remembered that (1) is the projection of the given straight line on zz, and (2) its projection on yz. Hence the straight line (3) is perpendicular to (1), and (4) is perpendicular to (2). .: A+ a = 0 and B+ 6 = 0 which are the equations of condition required. To find the equations to a straight line which passes through a given point, and is perpendicular to a given plane. Let the equation to the plane be 2 = Az + By + C The equations to the required line, sivce it passes through a point (2′, y′, z',) must be of the form z=a (s and, since it is perpendicular to the plane a = A, b = - - B. and therefore the equations required are 2 + A (= − 2) = 0, y-y+B (≈ — a) = 0. To find the distance (3) of the given point, in the last problem, from the plane. The equations to the straight line are z — ~ + A (≈ − x) = 0 .......................... (1) y −y + B (z − 2) = 0 ............................ (2) and the equation to the plane is 2 = Az + By + C which may be put under the form ......... (3) s — f = A (z — z) + B (y − y) + C + Az + By — 2' Now, if we suppose x, y, z, to be the co-ordinates of the point in which the perpendicular meets the plane, equations (1), (9), and (3), will hold good together, and we shall have But the distance (7) of the two points whose co-ordinates are x, y, z; L', I', %, is To find the distance (A) from a point in space to a straight line. Let the co-ordinates of the point be (x, y, z) and let the equations of the straight line be x = az + a ............... y = bz + ß ܐ )1( (2) The equation to a plane passing through the point x, y, z, and perpendicular to the given straight line, will be (z — z') + a (x − x') + b (y — y') = 0 (3) Now, if we suppose x, y, z; to be the co-ordinates of the point in which the plane meets the given straight line, the equations (1), (2), (3), will hold good together, and if we find values of (x — x'), (y — y'),(z — z'), from these equ tions and substitute the values thus obtained in the general expression for the distance of two given points in space, viz. r = √(x-x)' + (y — y')' + (« — 2)" we shall solve the problem. In order to effect this, let us put the equations (1) and (2) under the form (≈ — x') (1 + a2 + b2) + a (a — x' + az') + b (2~1 +62)=0 whence we find squaring these quantities and adding, we find No2 (1+a2 + b2) A= ·+(x —α)2+(y'—ß)2 + z22 — 2N. No = (≈′ — a)2 + (y — (3)2 + 222 - 1+ a2 +.b3• (x To determine the angle between two given planes. Let the equations to the planes be ............ .......................... z = Ax+ By + C ..... (1); % = Ax+ B'y + C ....... (2) If we let fall from the origin two straight lines perpendicular on these planes, the angle contained by the straight lines will be the same as the angle contained by the planes, let the equation to these straight lines be x = a'z and y = b'z the angle between them is known from the expression 1+ aa + bb' cos. @= √(1 + a2 + b2) (1 + a2 + b2) z'+a(x'—a)+b(y-3) 1 + a2 + b2 x = az } y = bx S But, in order that the straight lines may be perpendicular to the given planes, we must have and have and A+ a = 0, B + b = 0, A' + d = 0, B' + b = 0 Substituting therefore, the values of a, b, d, b', derived from these equations, we find that the expression of the cosine of the angle between the two planes, 1 + AA' + BB' cos. @= √(1 + A2 + B2) (1 + Aa + B') cos. (xy) = In order to find the angle which any plane makes with the co-ordinate planes, we have only to suppose that one of the above planes assumes in succession the position of the different co-ordinate planes, thus let us suppose, that (2) is the plane of xz, then its equation becomes y= 0, so that, A' 0, C = 0 and therefore, if we denote by the symbols (xz), (yz), (xy), the angles which the given plane makes with the planes xz, yz, xy, we have cos. (xz) = B cos. (yz) = 1 VI+A+B cos. (xy) + cos.2 (yz) + cos.2 (xz) = 1 cos. q = cos. (xz) cos. (x′z') + cos. (xy) cos. (x'y') + cos. (yz) cos. (y'z) To find the angle (4) contained by a plane and straight line in space. The angle sought is that which the straight line makes with its projection on the plane. If from any point in the given straight line we let fall a perpendi cular upon the plane, the angle contained between these two straight lines will be the complement of the required angle. Let the equation to the given plane be z = Ax + By + C The equation to the given straight line x = az + a ........... y = bx + ß ................................. (1) (2) (3) The equations of the line let fall perpendicular on the plane will be of the form x = dz + a y = b z + ß · (4) (5) But in order that this may be perpendicular to the given plane, we must have A+ a = 0 B+6=0 Now, the cosine of the angle contained by the two straight lines, is 1 + da + bb cos. @= √(1 + a2 + b2) (1 + a2 +62) It appears from what has been said above, that, in the present case = 90° ·é, and.. cos. Q = sin. 6. Substituting therefore for d, U, these value in terms of A and B, we find DIFFERENTIAL CALCULUS. CHAPTER L DEFINITIONS. In considering the relations which exist between different quantities, those which during the whole of any investigation are supposed to retain the same value are called constant quantities, those to which different values may be assigned are called variable quantities. Constant quantities are usually represented by the first letters of the alphabet, a, b, c, &c. variable quantities by the letters u, x, y, z, &c. When two or more variable quantities are connected in such a manner, that the value of one of them is determined by the value assigned to the other, the former is said to be a function of the other variables. Thus in the equation y= Ax+ Br2+C where the value of y depends upon the value assigned to x, y is said to be a function of x. In like manner if we have y = Az2 + Ba2 + Cx3 + D where the value of y depends upon the values assigned to ≈ and z, y is said to be a function of x and z. The words "function of r," are usually expressed by the symbols, ƒ (x), © (x), 4 (x), or similar abbreviations, and the above equations expressed in general terms would be written y = f(x) y = f(x, z) If y = ƒ (x), and a change takes place in the value of ƒ (x) such that x becomes x + h, x being quite indeterminate, and h any quantity whatever, either positive or negative, a corresponding change must take place in the value of y, which may then be represented by y'. If the quantity ƒ (x + h) be now developed in a series of the form f(x) + Ah + Bh2 + Ch3 + in which the first term is the original function f (x), and the other terms ascend regularly by positive and integral powers of h, and A, B, C, &c., are independent of h;* then the co-efficient of the simple power of h in this series is ... • We shall, in the mean time, take for granted that ƒ (x + h) can always be developed in a series of the above form, (showing, however, as we advance, that this is actually the case for all the parti. cular functions which fall under our notice) and defer the general demonstration of this principle until we proceed in Chapter V. to the discussion of Taylor's theorem. |