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.
#1 well (1,4,3)-(1,2,3) = (0,2,0)
#2 B'(x,y,z) = (x,y,-z)
#3 y = 6.1 sin55°
#4
Vx = 39 cos44°
in .9 sec, it travels .9 * Vx
Vy = 39 sin44°
h(t) = Vy*t - 16t^2
#5 Just like #3
oh my goodness, that was so helpful! Thankyou :) i think i got them all!
no
1. To find the image point after a translation, you need to add the components of the translation vector to the corresponding components of the original point. In this case, you have the translation vector (1, 2, 3) and the point (1, 4, 3).
Adding the x-components: 1 + 1 = 2
Adding the y-components: 4 + 2 = 6
Adding the z-components: 3 + 3 = 6
Therefore, after the translation, the point (1, 4, 3) would become (2, 6, 6).
2. To reflect a point over the xy-plane, you need to change the sign of the z-component of the original point. In this case, you have the point B(2, 1, 3).
Changing the sign of the z-component: -3
Therefore, the reflection of point B over the xy-plane would be B'(2, 1, -3).
3. To find the magnitude of the vertical component of a vector given its magnitude and direction, you can use the formula:
Vertical Component = Magnitude * sin(angle)
In this case, you have the magnitude of 6.1 inches and the direction of 55°.
Vertical Component = 6.1 * sin(55°)
Calculating the value using a calculator or trigonometric table, the magnitude of the vertical component would be approximately 5.06 inches.
4. To find how far the ball has traveled horizontally and vertically at a certain time, you can use the formulas for horizontal and vertical motion.
Horizontal Distance = Initial Velocity * Time * cos(angle)
In this case, you have an initial velocity of 39 feet per second, an angle of 44°, and a time of 0.9 seconds.
Horizontal Distance = 39 * 0.9 * cos(44°)
Calculating the value using a calculator or trigonometric table, the horizontal distance would be approximately 24.36 feet.
Vertical Distance = Initial Velocity * Time * sin(angle) - (1/2) * g * Time^2
In this case, you have an initial velocity of 39 feet per second, an angle of 44°, a time of 1.5 seconds, and the acceleration due to gravity g (usually taken as 32.2 feet per second squared).
Vertical Distance = 39 * 1.5 * sin(44°) - (1/2) * 32.2 * (1.5)^2
Calculating the value using a calculator or trigonometric table, the vertical distance would be approximately 22.48 feet.
28. To find the magnitude of the vertical component of a vector given its magnitude and direction, you can use the same formula as mentioned in question 3:
Vertical Component = Magnitude * sin(angle)
In this case, you have the magnitude of 5 inches and the direction of 32°.
Vertical Component = 5 * sin(32°)
Calculating the value using a calculator or trigonometric table, the magnitude of the vertical component would be approximately 2.67 inches.