What are some of the differences between the Ext1 and Ext2 topics of Vectors? Also what is the purpose of vector equations and why do we use them?
I can only give an elementary description (I'm not covering EVERY difference but I'll give a gist of it) of the differences;
Extension 1 topic only considers vectors in the second dimension i.e the cartesian plane (xy graph). It is a very fundamental introduction to the topic which goes over the addition/subtraction and scaling of vectors, how they can be represented in xi + yj form or a 2x1 matrice. You also learn about the dot product and its geometrical implications as well as projections. You also go into parametrics as well as touching vector equations. After this, you learn how this knowledge can be applied in physical contexts such as relative motion or systems in equilibrium.
Extension 2 goes further into vectors by extending it to the third dimension. A large component of this topic extends your pre-existing vectors knowledge from extension 1 such as the dot product, projection, representation of vectors into the 3D form. While it stops short of equations for planes, it does introduce you to parametric curves such as the helix or sphere which do come up in harder questions. You will gain more familiarity with vector equations in this topic as you have to deduce the nature of lines (Skewed, Orthogonal, Parallel etc.). You'll still receive vector questions in 2D but they will be more challenging than your Ext. 1 counterpart but the main focus is on vectors in 3D.
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The purpose of vector equations will become apparent when you do study these topics. What you said is pretty vague but all I'll say is that it is a better geometric representation of a line/plane than its common cartesian counterpart. What you'll find in the vectors topic is that you are able to more easily derive theorems or prove geometric properties which makes it extremely handy. In a sense its an intuitive way to move between algebra and geometry, like a bridge between both topics. You'll appreciate it more if you try doing regular problems in a non-vector approach.
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Vectors is a cool topic overall and given you grind it, you'll be more than fine with it. Very important you nail this topic down.
