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# lecture_30 - MA 265 LECTURE NOTES WEDNESDAY MARCH 26 Inner...

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MA 265 LECTURE NOTES: WEDNESDAY, MARCH 26 Inner Product Spaces Direction in R 2 and R 3 . We have seen that the angle between two vectors u and v in R 2 is given by θ = arccos u 1 v 1 + u 2 v 2 || u || || v || where 0 θ π. Similarly, the angle between two vectors u and v in R 2 is given by θ = arccos u 1 v 1 + u 2 v 2 + u 3 v 3 || u || || v || where 0 θ π. We use this to make a few of definitions: We say that two nonzero vectors u and v are in the same direction if θ = 0, i.e., 0 ; orthogonal or perpendicular if θ = π/ 2, i.e., 90 ; and in opposite directions if θ = π , i.e., 180 . Inner Products on R 2 and R 3 . There is a simple way to express the formula above. Define the standard inner product or dot product as the scalar u · v = ( u 1 v 1 + u 2 v 2 on R 2 ; u 1 v 1 + u 2 v 2 + u 3 v 3 on R 3 . Then we have the following formulas: The length of a vector is || v || = v · v . The distance from v to u is || v - u || = p ( v - u ) · ( v - u ) = p || u || 2 - 2 ( u · v ) + || v || 2 The angle between u and v is θ = arccos u · v || u || || v || . Hence it follows that two vectors u and v – not necessarily nonzero! – are in the same direction if u · v = || u || || v || ; orthogonal or perpendicular if u · v = 0; and in opposite directions if u · v = -|| u || || v || . Example. Consider the vector x = - 3 4 . We compute those vectors which are (1) in the same direction as x , (2) orthogonal to x , and (3) in the direction opposite to x . By looking at the geometry, we see that a vector in the same direction or opposite

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lecture_30 - MA 265 LECTURE NOTES WEDNESDAY MARCH 26 Inner...

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