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 find the image point after a translation using a vector, you can add the vector to the coordinates of the original point. In this case, we have the coordinates (1, 4, 3) and the translation vector (1, 2, 3).
To find the image point, you perform the following operation:
(1, 4, 3) + (1, 2, 3) = (1+1, 4+2, 3+3) = (2, 6, 6)
Therefore, the image point after the translation using the vector (1, 2, 3) is (2, 6, 6).
2. When reflecting a point over the xy-plane, you simply change the sign of the z-coordinate.
In this case, we have the coordinates of point B as (2, 1, 3). To find the reflection over the xy-plane, you change the sign of the z-coordinate to -3.
So, the coordinates of the reflection point B' would be (2, 1, -3).
3. To find the magnitude of the vertical component of a vector, you need to find the vertical component of the vector itself.
Given a vector with a magnitude of 6.1 inches and a direction of 55°, you can find the vertical component by using trigonometry.
The vertical component can be found using the formula: vertical component = magnitude * sin(angle)
= 6.1 * sin(55°)
To calculate this, you can use a scientific calculator or conversion tables to find the sine value of 55°.
Assuming the sine value is approximately 0.8192, you can calculate the vertical component as follows:
vertical component = 6.1 * 0.8192 = 4.99712
Thus, the magnitude of the vertical component of the vector is approximately 4.997 inches.
4. To find the horizontal distance traveled by a ball kicked at an angle, you can use the formula: horizontal distance = initial velocity * time * cos(angle)
In this case, the initial velocity is given as 39 feet per second and the angle is 44°.
After 0.9 seconds, the horizontal distance traveled would be:
horizontal distance = 39 * 0.9 * cos(44°)
To calculate this, you need to convert the angle from degrees to radians.
Assuming the cosine value of 44° is approximately 0.7193, you can calculate the horizontal distance as follows:
horizontal distance = 39 * 0.9 * 0.7193 = 24.3851 ft
So, after 0.9 seconds, the ball has traveled approximately 24.3851 feet horizontally.
Similarly, after 1.5 seconds, the horizontal distance traveled by the ball can be calculated using the same formula:
horizontal distance = 39 * 1.5 * cos(44°)
Assuming the cosine value of 44° is approximately 0.7193, you can calculate the horizontal distance as follows:
horizontal distance = 39 * 1.5 * 0.7193 = 39.0638 ft
So, after 1.5 seconds, the ball has traveled approximately 39.0638 feet horizontally.
5. The vertical component of a vector can be found using the formula: vertical component = magnitude * sin(angle).
Given a vector with a magnitude of 5 inches and a direction of 32°, you can find the magnitude of its vertical component.
The vertical component can be calculated using the formula mentioned above:
vertical component = 5 * sin(32°).
To calculate this, you can use a scientific calculator or conversion tables to find the sine value of 32°. Assuming the sine value is approximately 0.5299, you can calculate the vertical component as follows:
vertical component = 5 * 0.5299 = 2.6495.
Thus, the magnitude of the vertical component of the vector is approximately 2.6495 inches.