Find the direction of the following vector: u ⃗ = ⟨ − 10 , 10 ⟩ u ⃗=⟨-10,10⟩ u ⃗ = ⟨ − 10 , 10 ⟩ .

Let's see how we can solve this problem. So here we are asked to find the direction of the following vector v is equal to negative 110. Now to solve this, what I first like to do is kind of think about what this graph might look like or what this vector might look like graphically to see if I can get a general understanding of what direction the vector is going to be in. So if you kind of imagine we have some graph and I'll just draw this on the side here. You can think about this we'll have an x axis and a y axis, and if we're thinking about this vector negative ten ten, well, we know that negative ten on the x axis is gonna be somewhere back here, if we draw this to the origin, and then positive 10 on the y axis is gonna be somewhere up here. So our vector is gonna be somewhere over in this direction. Now, what this tells me is that if I can find this reference angle right here this angle that we make with the x axis, then I can find the total angle by using this angle that we reference, and that's really what we're looking for. Now to find this angle, recall that we can use this equation. The tangent of theta is equal to the y component divided by the x component. And I can see that I can have both the components given to us in this form. So we have that the tangent of theta is equal to the y component, which I can see is 10 divided by the x component, which I can see is negative 10. So So we have 10 divided by negative 10, and that all turns out to just be negative one. So what we end up with for theta is the inverse tangent of negative one, and the inverse tangent of negative one turns out to be negative 45 degrees. So what exactly does this tell us? Well, this tells us that the angle we're dealing with here is approximately equal to 45 degrees, right. We don't really have to worry about this negative sign right here, because we know that this is the angle that we're with reference to. And if this is the angle we make with this x axis, we can figure out this total angle by simply recognizing, well, this would be a 90 degree angle, that would be a 180 degrees total if we went all the way way around. So we see this subtract off that 45 degrees. So this angle is going to be a 180 degrees minus the 45 degrees that we just discovered and you could also think of it like we're just adding this negative 45 degree angle and a 180 minus 45 comes out to a 135. So your theta is going to be a 135 degrees. So the direction of our vector is a 135 degrees and that is the solution to this problem. So hope you found this video helpful. Thanks for watching.

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14. Vectors

Direction of a Vector

Precalculus

14. Vectors

Direction of a Vector

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Multiple Choice

Find the direction of the following vector: $u ⃗=⟨-10,10⟩$ .

45°

135°

−135°

315°

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Verified step by step guidance

To find the direction of a vector, we need to calculate the angle it makes with the positive x-axis. The vector given is $ \mathbf{u} = \langle -10, 10 \rangle $.

The direction angle $ \theta $ can be found using the tangent function: $ \tan(\theta) = \frac{y}{x} $, where $ x = -10 $ and $ y = 10 $.

Substitute the values into the tangent formula: $ \tan(\theta) = \frac{10}{-10} = -1 $.

To find $ \theta $, take the arctangent of $-1$. This gives $ \theta = \arctan(-1) $.

Since the vector is in the second quadrant (negative x and positive y), adjust the angle to reflect its position in the coordinate plane. The angle in the second quadrant is $ 180^\circ - \theta $.