Explanation Given information: The vectors, v = 2 i + j and w = − 4 i + 3 j Explanation: The decomposition of vector v into v 1 and v 2 , where v 1 is parallel to w , and v 2 is orthogonal to w , is v 1 = v ⋅ w | | w | | 2 w and v 2 = v − v 1 The vectors are v = 2 i + j and w = − 4 i + 3 j First find dot product of v and w . The dot product v ⋅ w is defined as the vector v ⋅ w = a 1 a 2 + b 1 b 2 , where v = a 1 i + b 1 j and w = a 2 i + b 2 j are two nonzero vectors in space. ⇒ v ⋅ w = ( 2 i + j ) ( − 4 i + 3 j ) = ( 2 ) ( − 4 ) + ( 1 ) ( 3 ) = − 8 + 3 = − 5 ⇒ v ⋅ w = − 5 The dot product of v and w is − 5 . Now to find the magnitude of each vector, The magnitude of v is defined by | | v | | = a 1 2 + b 1 2 where v = a 1 i + b 1 j

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Chapter 9, Problem 53RE

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The decomposition of vector v into two vectors one is parallel to w and other is orthogonal to w, where v=2i+j,w=−4i+3j.

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Chapter 9, Problem 52RE

Chapter 9, Problem 54RE

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The decomposition of vector v into two vectors one is parallel to w and other is orthogonal to w , where v = 2 i + j , w = − 4 i + 3 j .

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The decomposition of vector v into two vectors one is parallel to w and other is orthogonal to w, where v=2i+j,w=−4i+3j.

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