Consider [tex]$u =(5,-2)$[/tex] and [tex]$v =(10,-4)$[/tex]. What is the relationship between [tex]$u$[/tex] and [tex]$v$[/tex]?
The vectors point in the same direction.
The vectors form an acute angle.
The vectors form an obtuse angle.
The vectors point in opposite directions.
Answer (2)
Check if one vector is a scalar multiple of the other: v = k × u .
Find the scalar k such that ( 10 , − 4 ) = k × ( 5 , − 2 ) , which gives k = 2 .
Since k = 2 is a positive scalar, the vectors point in the same direction.
Alternatively, calculate the angle between the vectors using the dot product formula and verify that the angle is 0 degrees, confirming they point in the same direction. The vectors point in the same direction.
Explanation
Problem Analysis We are given two vectors u = ( 5 , − 2 ) and v = ( 10 , − 4 ) . Our goal is to determine the relationship between these vectors. The possible relationships are:
The vectors point in the same direction.
The vectors form an acute angle.
The vectors form an obtuse angle.
The vectors point in opposite directions.
Solution Plan One way to determine the relationship is to check if one vector is a scalar multiple of the other. If v = k × u for some positive scalar k , then the vectors point in the same direction. If v = k × u for some negative scalar k , then the vectors point in opposite directions. If the vectors are not scalar multiples, we can find the angle between them using the dot product formula.
Checking Scalar Multiple Let's check if v is a scalar multiple of u . We can write:
v = k × u
( 10 , − 4 ) = k × ( 5 , − 2 )
This gives us two equations:
10 = 5 k and − 4 = − 2 k
Solving for k in both equations:
k = 5 10 = 2 and k = − 2 − 4 = 2
Since k = 2 in both cases, v = 2 u .
Determining the Relationship Since v = 2 u and k = 2 is a positive scalar, the vectors u and v point in the same direction.
Alternative Method Using Dot Product Alternatively, we can calculate the dot product of u and v :
u ⋅ v = ( 5 ) ( 10 ) + ( − 2 ) ( − 4 ) = 50 + 8 = 58
Next, we calculate the magnitudes of u and v :
∣∣ u ∣∣ = 5 2 + ( − 2 ) 2 = 25 + 4 = 29
∣∣ v ∣∣ = 1 0 2 + ( − 4 ) 2 = 100 + 16 = 116 = 4 \t im es 29 = 2 29
Now, we can find the cosine of the angle between u and v :
cos ( θ ) = ∣∣ u ∣∣ \t im es ∣∣ v ∣∣ u ⋅ v = 29 \t im es 2 29 58 = 2 \t im es 29 58 = 58 58 = 1
Therefore, θ = arccos ( 1 ) = 0 degrees.
Conclusion Since the angle between the vectors is 0 degrees, the vectors point in the same direction.
Examples
Understanding the relationship between vectors is crucial in physics, especially when analyzing forces. For instance, if two forces acting on an object are represented by vectors u and v , and v = 2 u , it means the force v has twice the magnitude of force u and acts in the same direction, resulting in a stronger net force in that direction. This concept is fundamental in determining the motion and equilibrium of objects under multiple forces.
The vectors u = ( 5 , − 2 ) and v = ( 10 , − 4 ) point in the same direction because vector v is a positive scalar multiple of vector u (specifically, v = 2 u ). This is confirmed through both the scalar multiplication analysis and dot product calculation, which shows an angle of 0 degrees between them.
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