Find the horizontal and vertical components of the vector with the given length and direction, and write the vector in terms of the vectors i and j.
|v| = 24, θ = 30°
Well, let me take a funny turn and give you a clownish response! Brace yourself!
Alrighty then, let's get clowny with your question! So, we have a vector with a length of 24 and a direction of 30 degrees. Now, if I were a horizontal and vertical component of that vector, I'd call them "Holly" and "Verti"
Holly: Hey there Verti! Did you hear we've got a vector to break down?
Verti: Oh boy, what kind of mischief does this vector have in store for us?
Holly: Well, it's got a length of 24 and a direction of 30 degrees. Let's show it who's boss!
Verti: That sounds like a challenge! Let's start by finding our horizontal component. We'll call it "i" for insanity!
Holly: Nice one, Verti! To find the horizontal component, we multiply the length, 24, by the cosine of the direction, 30 degrees. So i = 24 * cos(30°).
Verti: And that turns out to be i = 24 * 0.866! We've got some numerical shenanigans going on!
Holly: Absolutely! Now it's time for you, Verti, to find our vertical component. Let's call it "j" for jubilation!
Verti: Oh, I'm always ready for some jubilation! To find the vertical component, we multiply the length, 24, by the sine of the direction, 30 degrees. So j = 24 * sin(30°).
Holly: You got it, Verti! That's j = 24 * 0.5. We've got some numerical clownery happening here!
Verti: Totally, Holly! Now, let's put it all together. The vector can be written as: 24 * 0.866 * i + 24 * 0.5 * j.
Holly: And that's how this crazy vector rolls, folks! We break it down, clown style!
Verti: Absolutely! So, ready or not, here comes the vector: 20.784i + 12j.
Holly: Boom! We've cracked the code and got ourselves a vector representation, clown-style!
Verti: Indeed, Holly! Mission accomplished! Now let's go grab some cotton candy and celebrate!
Holly: Sounds like a plan, Verti! Let's clown around some more after our tasty treats!
Verti: You got it, Holly! Clown Bot out!
(Note: The actual answer is 20.784i + 12j)
To find the horizontal and vertical components of the vector, we can use trigonometry.
Given:
Length of the vector, |v| = 24
Direction angle, θ = 30°
The horizontal component (x-component) can be found using cosine of the angle:
Horizontal component = |v| * cos(θ) = 24 * cos(30°)
The vertical component (y-component) can be found using sine of the angle:
Vertical component = |v| * sin(θ) = 24 * sin(30°)
Now, let's calculate the components:
Horizontal component = 24 * cos(30°)
Horizontal component ≈ 24 * 0.866
Horizontal component ≈ 20.784
Vertical component = 24 * sin(30°)
Vertical component ≈ 24 * 0.5
Vertical component ≈ 12
Now, let's write the vector in terms of the vectors i and j:
The vector can be written as:
Vector = Horizontal component * i + Vertical component * j
Vector = 20.784 * i + 12 * j
Therefore, the vector with the length of 24 and direction of 30 degrees can be written as 20.784i + 12j.
To find the horizontal and vertical components of a vector given its length and direction, we can use trigonometry.
Let's start by labeling the horizontal component as Vx and the vertical component as Vy. We can then use the following formulas:
Vx = |v| * cos(θ)
Vy = |v| * sin(θ)
Given that |v| = 24 and θ = 30°, we can substitute these values into the formulas:
Vx = 24 * cos(30°)
Vy = 24 * sin(30°)
To evaluate these expressions, we need to find the cosine and sine of 30°. Since 30° is a common angle, we can look up its cosine and sine values in a trigonometric table or use a calculator:
cos(30°) = √3/2 (approximately 0.866)
sin(30°) = 1/2 (approximately 0.5)
Now we can substitute these values into the expressions for Vx and Vy:
Vx = 24 * (0.866)
Vy = 24 * (0.5)
Simplifying these expressions, we get:
Vx = 20.784
Vy = 12.
Finally, we can write the vector in terms of the unit vectors i and j:
v = Vx * i + Vy * j
v = 20.784 * i + 12 * j
x: 24cos30° = 12√3
y: 24sin30° = 12