Position vector r⃗ has magnitude of 15.0 m and direction angle 250 ∘.
Find its components.
Find the components, magnitude, and direction of the vector −2r⃗ .
Where do I start?
Rx = 15 cos 250 = -5.13 m
Ry = 15 sin 250 = -14.1 m
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new one is - 2 times old
250 - 180 = 70 degrees is opposite
Rx = 30 cos 70 = 10.3
Ry = 30 sin 70 = 28.2
R^2 = (Rx^2+Ry)^2 = 30^2 (sin^2+cos^2) = 30^2
so it is 30 of course
direction is the old direction - 180 or 70
To find the components of a vector given its magnitude and direction angle, you can use trigonometry.
1. Finding the components of r⃗:
The direction angle, 250°, tells us that the vector makes an angle of 250° with the positive x-axis.
The x-component of the vector can be found using the formula: r⃗x = r⃗ * cos(θ)
Here, r⃗ is the magnitude of the vector, 15.0 m, and θ is the direction angle, 250°.
So, r⃗x = 15.0 m * cos(250°).
The y-component of the vector can be found using the formula: r⃗y = r⃗ * sin(θ)
Again, r⃗ is the magnitude of the vector, 15.0 m, and θ is the direction angle, 250°.
So, r⃗y = 15.0 m * sin(250°).
2. Finding the components, magnitude, and direction of −2r⃗:
To find the components of −2r⃗, we need to find the components of r⃗ first and then multiply them by −2.
The x-component of −2r⃗ is given by: −2r⃗x = -2 * r⃗x
Similarly, the y-component of −2r⃗ is given by: −2r⃗y = -2 * r⃗y
The magnitude of −2r⃗ can be found using the formula: |−2r⃗| = |-2| * |r⃗|
Here, |r⃗| is the magnitude of r⃗.
So, |−2r⃗| = |-2| * 15.0 m.
The direction of −2r⃗ will be the same as the direction of r⃗, but the magnitude will be doubled.
I hope this helps! Let me know if you have any further questions.