Let [tex]\mathbf{u} = \overrightarrow{XY}[/tex] where X = (-4, 6) and Y = (8, 3). What is the magnitude of 2[tex]\mathbf{u}[/tex]?
Answer (2)
The magnitude of the vector 2 u is approximately 24.74, derived from first finding the vector u and then scaling it. The initial calculation showed that ∣∣ u ∣∣ is approximately 12.37. By multiplying this magnitude by 2, we obtain the result for 2 u .
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To find the magnitude of 2 u where u = X Y , we'll follow these steps.
Calculate the vector u : u = X Y = ( x 2 − x 1 , y 2 − y 1 ) , where X = ( − 4 , 6 ) and Y = ( 8 , 3 ) .
u = ( 8 − ( − 4 ) , 3 − 6 ) = ( 8 + 4 , 3 − 6 ) = ( 12 , − 3 )
Calculate the magnitude of u : The magnitude ∣∣ u ∣∣ of a vector u = ( a , b ) is given by a 2 + b 2 .
∣∣ u ∣∣ = 1 2 2 + ( − 3 ) 2 = 144 + 9 = 153
Find the vector 2 u : 2 u = 2 × ( 12 , − 3 ) = ( 24 , − 6 )
Calculate the magnitude of 2 u : The magnitude ∣∣2 u ∣∣ is given by:
∣∣2 u ∣∣ = 2 4 2 + ( − 6 ) 2 = 576 + 36 = 612
Simplify the magnitude: To simplify, notice:
612 = 4 × 153 = 4 × 153 = 2 153
Therefore, the magnitude of 2 u is 2 153 .
