What is the magnitude of vector [tex]v[/tex], with initial point [tex](2,-7)[/tex] and terminal point [tex](-4,5)[/tex]?
Answer (2)
Calculate the components of the vector v by subtracting the initial point from the terminal point: v = ( − 4 − 2 , 5 − ( − 7 )) = ( − 6 , 12 ) .
Calculate the magnitude of the vector v using the formula ∣∣ v ∣∣ = ( − 6 ) 2 + ( 12 ) 2 .
Simplify the expression: ∣∣ v ∣∣ = 36 + 144 = 180 .
Express the final answer in simplest radical form: 6 5 .
Explanation
Problem Analysis We are given a vector v with initial point ( 2 , − 7 ) and terminal point ( − 4 , 5 ) . Our goal is to find the magnitude of this vector.
Calculate Vector Components First, we need to find the components of the vector v . We do this by subtracting the coordinates of the initial point from the coordinates of the terminal point. So, if the initial point is ( x 1 , y 1 ) and the terminal point is ( x 2 , y 2 ) , the components of the vector are ( x 2 − x 1 , y 2 − y 1 ) . In our case, ( x 1 , y 1 ) = ( 2 , − 7 ) and ( x 2 , y 2 ) = ( − 4 , 5 ) . Therefore, the components of vector v are: ( − 4 − 2 , 5 − ( − 7 )) = ( − 6 , 12 ) So, v = ( − 6 , 12 ) .
Calculate Magnitude of Vector Next, we need to find the magnitude of the vector v . The magnitude of a vector ( a , b ) is given by the formula ∣∣ v ∣∣ = a 2 + b 2 . In our case, v = ( − 6 , 12 ) , so a = − 6 and b = 12 . Therefore, the magnitude of vector v is:
∣∣ v ∣∣ = ( − 6 ) 2 + ( 12 ) 2 = 36 + 144 = 180 Now, we simplify the square root: 180 = 36 × 5 = 36 × 5 = 6 5 Therefore, the magnitude of vector v is 6 5 .
Final Answer The magnitude of vector v with initial point ( 2 , − 7 ) and terminal point ( − 4 , 5 ) is 6 5 .
Examples
Vectors are used extensively in physics to represent quantities that have both magnitude and direction, such as velocity, force, and displacement. For example, if you're analyzing the motion of a car, you can represent its displacement as a vector. Knowing the initial and final positions, you can calculate the displacement vector and its magnitude, which tells you the total distance the car traveled in a straight line from its starting point, regardless of the path it took. This is crucial for understanding the car's overall change in position.
To find the magnitude of vector v , we calculate its components from the given points and then apply the magnitude formula. The components are ( − 6 , 12 ) , leading to a magnitude of 6 5 . Therefore, the answer is 6 5 .
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