1. $(-5,6)(-3,3)$
a. Draw the Coordinates
b. Solve the distance
c. Solve the midpoint
d. From the origin connect the midpoint
e. Solve the vector.
f. Solve the angle of Direction
Answer (1)
Calculate the distance between points A(-5, 6) and B(-3, 3) using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 = 13 ≈ 3.61 .
Determine the midpoint M of the line segment AB using the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) = ( − 4 , 4.5 ) .
Find the vector A B from A to B by subtracting the coordinates: A B = ( x 2 − x 1 , y 2 − y 1 ) = ( 2 , − 3 ) .
Calculate the direction angle θ of the vector A B using θ = arctan ( x 2 − x 1 y 2 − y 1 ) , adjusting for the correct quadrant: θ ≈ 303.6 9 ∘ .
Explanation
Problem Analysis We are given two points A(-5, 6) and B(-3, 3). We need to find the distance between them, the midpoint of the line segment connecting them, the vector from A to B, and the direction angle of the vector.
Calculate the Distance To find the distance d between points A and B, we use the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Substituting the coordinates of A and B: d = ( − 3 − ( − 5 ) ) 2 + ( 3 − 6 ) 2 = ( 2 ) 2 + ( − 3 ) 2 = 4 + 9 = 13 ≈ 3.61 So, the distance between A and B is approximately 3.61 units.
Calculate the Midpoint To find the midpoint M of the line segment AB, we use the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Substituting the coordinates of A and B: M = ( 2 − 5 + ( − 3 ) , 2 6 + 3 ) = ( 2 − 8 , 2 9 ) = ( − 4 , 4.5 ) So, the midpoint of the line segment AB is (-4, 4.5).
Calculate the Vector To find the vector A B from A to B, we subtract the coordinates of A from the coordinates of B: A B = ( x 2 − x 1 , y 2 − y 1 ) Substituting the coordinates of A and B: A B = ( − 3 − ( − 5 ) , 3 − 6 ) = ( 2 , − 3 ) So, the vector from A to B is (2, -3).
Calculate the Angle of Direction To find the direction angle θ of the vector A B , we use the formula: θ = arctan ( x 2 − x 1 y 2 − y 1 ) Substituting the components of A B :
θ = arctan ( 2 − 3 ) Since the vector is in the fourth quadrant (x is positive and y is negative), we need to add 360 degrees to the angle to get the correct direction angle. θ = arctan ( 2 − 3 ) + 36 0 ∘ ≈ − 56.3 1 ∘ + 36 0 ∘ = 303.6 9 ∘ So, the direction angle of the vector from A to B is approximately 303.69 degrees.
Final Answer The distance between the points (-5, 6) and (-3, 3) is approximately 3.61 units. The midpoint of the line segment connecting these points is (-4, 4.5). The vector from (-5, 6) to (-3, 3) is (2, -3). The direction angle of this vector is approximately 303.69 degrees.
Examples
Understanding distance, midpoint, vectors, and angles is crucial in navigation. For example, a GPS system uses coordinates to locate positions, calculates the distance between two points to estimate travel time, and determines the direction angle to guide movement. These calculations are fundamental in mapping applications and route planning, making navigation efficient and accurate.
