A vector has an initial point at $(3,-5)$ and a terminal point at $(-2,3)$. What are the characteristics of this vector?
A. The magnitude is $\sqrt{13}$, and the direction angle is about $58^{\circ}$.
B. The magnitude is $\sqrt{13}$, and the direction angle is about $122^{\circ}$.
C. The magnitude is $\sqrt{89}$, and the direction angle is about $58^{\circ}$.
D. The magnitude is $\sqrt{89}$, and the direction angle is about $122^{\circ}$.
Answer (2)
Calculate the vector components by subtracting the initial point coordinates from the terminal point coordinates: ( − 5 , 8 ) .
Determine the magnitude of the vector using the formula: ( − 5 ) 2 + 8 2 = 89 .
Calculate the direction angle using the arctangent function, considering the quadrant: arctan ( − 5 8 ) ≈ 12 2 ∘ .
The vector has magnitude 89 and direction angle approximately 12 2 ∘ .
The magnitude is 89 , and the direction angle is about 12 2 ∘ .
Explanation
Define initial and terminal points Let A = ( 3 , − 5 ) be the initial point and B = ( − 2 , 3 ) be the terminal point of the vector. We want to find the magnitude and direction angle of the vector A B .
Calculate vector components The components of the vector A B are given by the difference between the coordinates of the terminal point and the initial point: ( − 2 − 3 , 3 − ( − 5 )) = ( − 5 , 8 ) So, the vector is A B = ( − 5 , 8 ) .
Calculate the magnitude The magnitude of the vector A B is given by the formula: ∥ A B ∥ = ( − 5 ) 2 + 8 2 = 25 + 64 = 89 ≈ 9.43 Thus, the magnitude of the vector is 89 .
Calculate the direction angle The direction angle θ is given by tan − 1 ( − 5 8 ) . Since the x-component is negative and the y-component is positive, the angle lies in the second quadrant. We can use the atan2 function to find the correct angle. θ = arctan ( − 5 8 ) ≈ 122.0 1 ∘ Thus, the direction angle is approximately 12 2 ∘ .
Compare with given options Comparing our results with the given options, we find that the magnitude is 89 and the direction angle is about 12 2 ∘ .
Examples
Vectors are used extensively in physics to represent forces, velocities, and accelerations. For example, if you are pushing a box across the floor, the force you apply can be represented as a vector with a magnitude (how hard you are pushing) and a direction (the angle at which you are pushing). Understanding the magnitude and direction of these vectors is crucial for calculating the net force and predicting the motion of the box.
The vector from the point (3, -5) to (-2, 3) has a magnitude of 89 and a direction angle of approximately 12 2 ∘ . Therefore, the correct answer is option D.
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