What is the magnitude of vector $v$ with initial point $(2,-7)$ and terminal point $(-4,5)$?
Answer (1)
Find the components of the vector v by subtracting the initial point from the terminal point: v = ( − 6 , 12 ) .
Calculate the magnitude of the vector using the formula ∣∣ v ∣∣ = a 2 + b 2 .
Substitute the components into the formula: ∣∣ v ∣∣ = ( − 6 ) 2 + ( 12 ) 2 = 180 .
Simplify the square root to find the magnitude: 6 5 .
Explanation
Problem Analysis We are given the initial point ( 2 , − 7 ) and the terminal point ( − 4 , 5 ) of vector v . Our goal is to find the magnitude of this vector.
Finding Vector Components First, we need to find the components of the vector v . The components are found by subtracting the coordinates of the initial point from the coordinates of the terminal point. So, we have:
v = ( x 2 − x 1 , y 2 − y 1 ) = ( − 4 − 2 , 5 − ( − 7 )) = ( − 6 , 12 )
Calculating Magnitude Now that we have the components of the vector, we can find its magnitude. The magnitude of a vector v = ( a , b ) is given by the formula:
∣∣ v ∣∣ = a 2 + b 2
In our case, a = − 6 and b = 12 . So, the magnitude of vector v is:
∣∣ v ∣∣ = ( − 6 ) 2 + ( 12 ) 2 = 36 + 144 = 180
Simplifying the Result We can simplify the square root:
180 = 36 ⋅ 5 = 36 ⋅ 5 = 6 5
Final Answer Therefore, the magnitude of vector v is 6 5 .
Examples
Vectors are used extensively in physics to represent forces and velocities. For example, if you are pushing a box with a certain force at an angle, you can represent that force as a vector. The magnitude of the vector would represent the strength of the force, and the direction of the vector would represent the direction in which you are pushing. Knowing the magnitude and direction allows you to calculate the effect of the force on the box's motion.
