Find the magnitude and direction of CD for the given coordinates. Round to the nearest tenth. C(−4, 1), D(−7, 3)
mag = √[(-7 - -4)^2 + (3 - 1)^2]
tan(Θ) = (3 - 1) / (-7 - -4) ... Quad II
To find the magnitude and direction of CD, we first need to determine the vector CD. The vector CD is given by the difference between the coordinates of point D and point C.
CD = (x₂ - x₁, y₂ - y₁)
where (x₁, y₁) are the coordinates of point C and (x₂, y₂) are the coordinates of point D.
Given C(-4, 1) and D(-7, 3), we can compute:
CD = (-7 - (-4), 3 - 1)
= (-3, 2)
The magnitude of a vector is given by the formula:
|CD| = √(x² + y²)
where (x, y) are the components of the vector.
Using the coordinates calculated above, we can calculate the magnitude as follows:
|CD| = √((-3)² + 2²)
= √(9 + 4)
= √13
≈ 3.6 (rounded to the nearest tenth)
To find the direction, we can calculate the angle made by the vector CD with the positive x-axis. This can be done using the arctan function.
θ = arctan(y/x)
where x and y are the components of the vector.
In this case:
θ = arctan(2/(-3))
≈ -33.7 degrees (rounded to the nearest tenth)
So, the magnitude of CD is approximately 3.6 and the direction is approximately -33.7 degrees.
To find the magnitude and direction of CD, we can use the distance formula and trigonometry.
First, let's find the distance between points C(-4, 1) and D(-7, 3):
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the values, we get:
Distance = √[(-7 - (-4))^2 + (3 - 1)^2]
= √[(-7 + 4)^2 + (3 - 1)^2]
= √[(-3)^2 + (2)^2]
= √[9 + 4]
= √13
≈ 3.6 (rounded to the nearest tenth)
The magnitude of CD is approximately 3.6.
Now, let's find the direction of CD. We can use trigonometry to find the angle between CD and the positive x-axis.
First, let's find the horizontal and vertical distances between the points:
Horizontal distance = x2 - x1 = -7 - (-4) = -3
Vertical distance = y2 - y1 = 3 - 1 = 2
To find the angle, we can use the arctan function:
Angle = arctan(vertical distance / horizontal distance)
Substituting the values, we get:
Angle = arctan(2 / -3)
≈ -33.7 degrees (rounded to the nearest tenth)
The direction of CD is approximately -33.7 degrees.
Therefore, the magnitude of CD is approximately 3.6 and the direction is approximately -33.7 degrees.