What is the relationship between ‖v‖ and v·v ?

‖v‖ and v·v are related as follows v·v=‖v‖2

Q .8. - Calculus | StudyQuestionHub

$What is the relationship between ‖ v ‖ and v \cdot v ?$

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Answer

$‖ v ‖ and v \cdot v are related as follows v \cdot v = ‖ v ‖^{2}$

1Step 1:Given information

$‖ v ‖ and v \cdot v$

2Step 2:Simplification

$Consider the vector ‖ v ‖ and v \cdot v .$

$If u and v are the vector such that u = <u_{1}, u_{2}, u_{3}> and v = <v_{1}, v_{2}, v_{3}>, then dot product is given$

$by u . v = u_{1} v_{1} + u_{2} v_{2} + w_{1} w_{2} .$

$NOW, v = <v_{1}, v_{2}, v_{3}>$

$Therefore,$

$V . V = v_{1} \cdot v_{1} + v_{2} \cdot v_{2} + v_{3} \cdot v_{3}$

$= v_{1}^{2} + v_{2}^{2} + v_{2}^{2}$

$Therefore, the dot product of the two vectors v.v is v_{1}^{2} + v_{2}^{2} + v_{2}^{2}$

$If v is the vector in R^{3} such that v = <v_{1}, v_{2}, v_{3}>, then norm of the vector is given by$

$‖ v ‖ = \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}$

From (1) and (2) it is clear that

$V \cdot V = ‖ V ‖^{2}$

$Hence, ‖ v ‖ and v \cdot v are related as follows v \cdot v = ‖ v ‖^{2}$