1. A triangular prism has vertices at A(2, 0, 0), B(2, 1, 3), C(2, 2, 0), D(0, 0, 0), E(0, 1, 3), and F(0, 2, 0).
Which image point has the coordinates (1, 4, 3) after a translation using the vector 1, 2, 3?
2. What point represents a reflection of B over the xy-plane?
B'(?)
3. The vector has a magnitude of 6.1 inches and a direction of 55°. Find the magnitude of its veritcal component. (?) inches
4. Aaron kicked a soccer ball with an initial velocity of 39 feet per second at an angle of 44° with the horizontal.
After 0.9 second, how far has the ball traveled horizontally? (?)ft
After 1.5 seconds, how far has the ball traveled vertically? (?) ft
28. The vector has a magnitude of 5 inches and a direction of 32°. Find the magnitude of its vertical component. ?in.
See previous post.
1. To perform a translation on a point, you need to add the components of the translation vector to the coordinates of the original point. In this case, you need to add the components (1, 2, 3) to the coordinates of the point (1, 4, 3).
To find the image point, you add the corresponding components:
x-coordinate of the image point = x-coordinate of the original point + x-component of the translation vector = 1 + 1 = 2
y-coordinate of the image point = y-coordinate of the original point + y-component of the translation vector = 4 + 2 = 6
z-coordinate of the image point = z-coordinate of the original point + z-component of the translation vector = 3 + 3 = 6
Therefore, the image point with the coordinates (1, 4, 3) after the translation using the vector (1, 2, 3) is (2, 6, 6).
2. To reflect a point over the xy-plane, you need to keep the x and y coordinates the same, but change the sign of the z-coordinate.
To find the reflection of point B(x, y, z) over the xy-plane, the coordinates of the reflected point B' will be (x, y, -z).
Therefore, the reflection of point B over the xy-plane is B'(2, 1, -3).
3. To find the vertical component of a vector given its magnitude and direction, you can use trigonometry.
The vertical component of a vector can be found by multiplying the magnitude of the vector by the sine of the angle between the vector and the vertical axis.
Vertical component = magnitude of the vector * sin(angle)
In this case, the magnitude of the vector is 6.1 inches and the angle is 55°.
Vertical component = 6.1 inches * sin(55°)
Using a calculator, you can evaluate the sine of 55° and multiply it by 6.1 inches to find the magnitude of the vertical component.
4. To find the horizontal and vertical distances traveled by a projectile, you can use the equations of motion.
For the horizontal distance:
Horizontal distance = initial velocity * time * cosine(angle)
In this case, the initial velocity is 39 feet per second, the time is 0.9 seconds, and the angle is 44°.
Horizontal distance = 39 feet/second * 0.9 seconds * cos(44°)
Using a calculator, you can evaluate the cosine of 44° and multiply it by 39 feet per second and 0.9 seconds to find the horizontal distance.
For the vertical distance (assuming no air resistance):
Vertical distance = initial velocity * time * sine(angle) - (1/2) * acceleration * time^2
In this case, the initial velocity is 39 feet per second, the time is 1.5 seconds, the angle is 44°, and the acceleration due to gravity is approximately 32.2 feet per second squared.
Vertical distance = 39 feet/second * 1.5 seconds * sin(44°) - (1/2) * 32.2 feet/second^2 * (1.5 seconds)^2
Using a calculator, you can evaluate the sine of 44°, square the time, and multiply by 32.2 feet per second squared to find the vertical distance.
28. To find the vertical component of a vector given its magnitude and direction, you can use trigonometry.
The vertical component of a vector can be found by multiplying the magnitude of the vector by the sine of the angle between the vector and the vertical axis.
Vertical component = magnitude of the vector * sin(angle)
In this case, the magnitude of the vector is 5 inches and the angle is 32°.
Vertical component = 5 inches * sin(32°)
Using a calculator, you can evaluate the sine of 32° and multiply it by 5 inches to find the magnitude of the vertical component.