By Philip Franklin

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**Extra resources for Methods of Advanced Calculus 1ST Edition**

**Example text**

Now b' == ba'/ a and c' == ca]« imply that a' : b' : c' == 1 : bfa : cia == a: b: c. This means that the planes 1f and 1f' are parallel but not coincident if and only if a' : b' : c' == a : b : c, but a' : b' : c' : d' =1= a : b : c : d. 3), suppose (without loss of generality) that the coefficient of y is non-zero; then we have y == Az + B and x == Cz + D where A, B, C, Dare constants, and these two equations define a straight line. To summarise what we have found here: the two planes 1f and n' intersect in a straight line unless (i) they are parallel, in which case a' : b' : c' == a : b : c, a' : b' : c' : d' =1= a : b : c : d, or (ii) they are coincident, in which case a' : b' : c' : d' == a : b : c : d.

In general the clearest images are obtained when the rays meeting the screen are as close to being perpendicular to the screen as possible. When showing slides it is usual to have the line joining the centre of the picture on the screen to the lens of the projector at right angles to the screen. 3 26 Vectors in 2 or 3 Dimensions distortion a flat screen would produce, curved screens are used, and the projector is placed on the axis of the cylinder in which the screen lies, so that all the horizontal rays from the projector meet the screen orthogonally.

Definition 5 The vectors atVI are said to be linearly dependent if we can find scalars not all zero, such that VI, V2, ... ,Vn aI, a2, ... , an + a2V2 + ... + anVn == O. Remember that it is the zero vector on the right.