Prove the following relationship using the cosine rule: ab=| a || b | cos 0, where...

  

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Prove the following relationship using the cosine rule: ab=| a || b | cos 0, where is the angle between vectors a and b. The dot product can be extended to an arbitrary number of dimensions: N a. b =ajbj J=1 The relationship between dot product and cosine also holds in three and more dimensions. Argue that this suggests that the dot product is invariant under rotations (this is in fact true). The dot product is very remarkable: it relates two bags of numbers: (the components of the two vec- tors; numbers that can be stored, read from file etc.) via a simple algebraic formula to a geometrical concept: the projection from one vector onto another (Fig. 3). A A cose B Figure 3: Projection of one vector onto another via the scalar product Find an algebraic expression that relates the components of two vectors that are perpendicular. Prove the following relationship using the cosine rule: ab=| a || b | cos 0, where is the angle between vectors a and b. The dot product can be extended to an arbitrary number of dimensions: N a. b =ajbj J=1 The relationship between dot product and cosine also holds in three and more dimensions. Argue that this suggests that the dot product is invariant under rotations (this is in fact true). The dot product is very remarkable: it relates two bags of numbers: (the components of the two vec- tors; numbers that can be stored, read from file etc.) via a simple algebraic formula to a geometrical concept: the projection from one vector onto another (Fig. 3). A A cose B Figure 3: Projection of one vector onto another via the scalar product Find an algebraic expression that relates the components of two vectors that are perpendicular.