Fundamental theorems of vectors Assignment Help

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Fundamental theorems of vectors:

Fundamental theorem of vectors in two-dimensions

If 2104_Addition of two vectors9.png and 1830_Addition of two vectors10.pngbe 2 non-zero non-collinear vectors, then the vector 1900_Fundamental theorems of vectors1.png in the plane of 2104_Addition of two vectors9.pngand 1830_Addition of two vectors10.png can be expressed uniquely as a linear combination of 2104_Addition of two vectors9.pngand 1830_Addition of two vectors10.png that is there exist unique l, mÎR such that l2104_Addition of two vectors9.png +m1830_Addition of two vectors10.png = 1900_Fundamental theorems of vectors1.png.

This means that if l12104_Addition of two vectors9.png + m11830_Addition of two vectors10.png = l22104_Addition of two vectors9.png + m21830_Addition of two vectors10.png then l1 = l2 and m1 = m2.

Fundamental theorem of vectors in three-dimensions

If 2104_Addition of two vectors9.png1830_Addition of two vectors10.pngand 808_Fundamental theorems of vectors.png be 3 non-zero, non-coplanar vectors in space, then any vector 1900_Fundamental theorems of vectors1.png in space can be expressed uniquely as linear combination of 2104_Addition of two vectors9.png1830_Addition of two vectors10.pngand 1900_Fundamental theorems of vectors1.png.

That is there exist unique l, m, n ÎR such that l2104_Addition of two vectors9.png + m1830_Addition of two vectors10.png + n 808_Fundamental theorems of vectors.png = 1900_Fundamental theorems of vectors1.png

This means that if l12104_Addition of two vectors9.png + m11830_Addition of two vectors10.png+ n1808_Fundamental theorems of vectors.png = l22104_Addition of two vectors9.png+ m21830_Addition of two vectors10.png+ n2808_Fundamental theorems of vectors.png, then l1 = l2, m1 = m2 and n1 = n2.

 

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