If a vector has magnitude ∣v⃗∣=5|v ⃗ |=5∣v⃗∣=5 and direction θ=7π4\theta=\frac{7\pi}{4}θ=47π, find the vector’s horizontal and vertical components.

v x = 3.54 v_{x}=3.54 v x = 3.54 and v y = − 3.54 v_{y}=-3.54 v y = − 3.54

If a vector has magnitude ∣ v ⃗ ∣ = 5 |v ⃗ |=5 ∣ v ⃗∣ = 5 and direction θ = 7 π 4 \theta=\frac{7\pi}{4} θ = 4 7 π , find the vector’s horizontal and vertical components.

Let's see how we can solve this problem. So, here we're told if a vector v has a magnitude of 5, and the direction of 7pi over 4 radians, find the vectors horizontal and vertical components. Okay. Now, whenever I'm solving these types of problems, I always like to start with a graph. So if we draw a simple x y graph over here, we can get a general sense of what our vector looks like. Now I can see that our vector has a magnitude of 5, but the direction is 7pi over 4. Now 7pi over 4 radians is equivalent to about 315 degrees, so you're going to be all the way over here with your vector. So the vector 5, the with a magnitude of 5 is gonna be right about there, and our total angle or direction here is 315 degrees, which is also written as 7pi over 4 radians. So this is what our vector ends up looking like. And if we want to calculate the x component, we can do that with the equation that we're getting. We can also do this for the y component. So what I'm first going to do is find the x component using this equation. We're gonna have the magnitude of v multiplied by the cosine of the angle. Now the magnitude of our vector is 5, and we'll have that times the cosine of 7 pi over 4. Now make sure your calculator is in radian mode when calculating this because we're now dealing with radians. So if you plug this in, you should get a result of approximately 3.54 and that's going to be the x component of our vector. But now let's see if we can find the y component. The y component is going to be the magnitude of our vector multiplied by the sine of the angle. So we're going to have 5 times the sine of 7 pi over 4. Now plugging this into a calculator, your answer should come out to approximately negative 3.54. So this right here is our y component and that is how you can find the horizontal and vertical components of this vector. So hope you found this video helpful. Thanks for watching.

If a vector has magnitude ∣ v ⃗ ∣ = 5 |v ⃗ |=5 ∣ v ⃗∣ = 5 and direction θ = 7 π 4 \theta=\frac{7\pi}{4} θ = 4 7 π , find the vector’s horizontal and vertical components.

v x = 3.54 v_{x}=3.54 v x = 3.54 and v y = − 3.54 v_{y}=-3.54 v y = − 3.54

v x = 4.98 v_{x}=4.98 v x = 4.98 and v y = 0.479 v_{y}=0.479 v y = 0.479

v x = 0.479 v_{x}=0.479 v x = 0.479 and v y = 4.98 v_{y}=4.98 v y = 4.98

v x = − 3.54 v_{x}=-3.54 v x = − 3.54 and v y = 3.54 v_{y}=3.54 v y = 3.54

8. Vectors

Direction of a Vector

Trigonometry

8. Vectors

Direction of a Vector

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Multiple Choice

If a vector has magnitude $|v ⃗ |=5$ and direction $\theta=\frac{7\pi}{4}$ , find the vector’s horizontal and vertical components.

$v_{x}=4.98$ and $v_{y}=0.479$

$v_{x}=0.479$ and $v_{y}=4.98$

$v_{x}=-3.54$ and $v_{y}=3.54$

$v_{x}=3.54$ and $v_{y}=-3.54$

Verified step by step guidance

To find the horizontal and vertical components of a vector given its magnitude and direction, we use the formulas: $ v_x = |\vec{v}| \cos(\theta) $ and $ v_y = |\vec{v}| \sin(\theta) $.

Substitute the given magnitude $ |\vec{v}| = 5 $ and direction $ \theta = \frac{7\pi}{4} $ into the formulas.

Calculate the horizontal component: $ v_x = 5 \cos\left(\frac{7\pi}{4}\right) $.

Calculate the vertical component: $ v_y = 5 \sin\left(\frac{7\pi}{4}\right) $.

Evaluate the trigonometric functions: $ \cos\left(\frac{7\pi}{4}\right) $ and $ \sin\left(\frac{7\pi}{4}\right) $ to find the exact values of $ v_x $ and $ v_y $.

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