The figure shows a small plane flying at a speed of 180 miles per hour on a bearing of N 50^∘ E

step 1 Hello, hope you're doing well. So for this problem, we have our graph, and we have our plane tha

The figure shows a small plane flying at a speed of 180...

Key Concepts

Vector Representation

This concept involves describing physical quantities such as velocity or force using both magnitude and direction. In two-dimensional motion problems, vectors are often represented graphically and analytically, with one component along the horizontal (x-axis) and one along the vertical (y-axis), or using polar notation specifying an angle relative to a reference direction.

Conversion between Bearings and Standard Angles

Bearings are navigational directions given relative to the north (or south) typically measured clockwise, while standard angles in mathematics are usually measured counterclockwise from the positive x-axis (east). Understanding how to convert between these systems is essential to correctly interpreting and solving navigation and vector problems.

Decomposition of Vectors into Components

To solve problems involving vectors, it is common to break a vector into its horizontal and vertical components using trigonometric functions such as sine and cosine. This decomposition allows for easier vector addition and is essential in analyzing forces and velocities in multiple dimensions.

Vector Addition

This concept involves combining two or more vectors to find a resultant vector. By adding the corresponding components of each vector, one can determine the overall effect, such as the total velocity when combining the motion of a plane with the wind's influence.

Magnitude Calculation

After obtaining the components of a vector (or resultant vector), the magnitude or length is calculated using the Pythagorean theorem. This is critical when finding the ground speed of an object, which is the resultant speed when vector components are combined.

Direction Angle Determination

Once the resultant vector's components are known, trigonometric functions, typically the inverse tangent function, are used to determine the direction angle of the resultant vector. This angle is then often converted to a navigational bearing to provide meaningful directional information in a real-world context.

Transcript

00:01 Hello, hope you're doing well.

00:03 So for this problem, we have our graph, and we have our plane that's flying in that direction.

00:11 This is our vector v.

00:13 And the problem tells us that the angle between north and, or the angle between this v vector and the y axis, the north direction here is 50 degrees.

00:26 It's our v vector.

00:28 And then we're told that our w vector is just going straight east.

00:32 That.

00:34 Okay, so in order to figure out the vector components of v and w, so for expressing these vectors in terms of their x and y components, the x component of our v vector is going to be our v times sine 50 degrees.

00:56 So that's going to be the v component that's in this direction, the x direction.

01:02 And then to get the the y components of our vector v.

01:08 We're going to multiply v by cosine of 50 degrees.

01:13 That's going to give the component of the vector v, i'm sorry, that's in this direction, the vertical direction.

01:22 So this is our x component right here, and this is our y component right here.

01:27 So the final result for the vector of v in terms of its x and y components would be v, sign 50 degrees i plus v cosine 50 degrees j so that's and then our w vector would just be whatever the value of w is in the i direction because it's just um in the x direction so now once you have um so that kind of would help us out with part a so in order to find result in vector between v and w to add two vectors together, you just add their x components and y components together.

02:09 So w is just going to be wi.

02:11 So you add these two components together to get some number i, plus you add the j components together.

02:19 In this case, there's no j component for your w vectors.

02:22 You just keep this v cosine theta or v .cosin 53s in j.

02:27 So you just add the components of vectors when you add two vectors.

02:31 And then in order to find the magnitude of two, vectors, or the magnitude of a vector, let's say you have a vector that's vx in the i direction plus v y in the j direction.

02:44 Find the magnitude of the vector v.

02:48 So that's essentially the length of the vector, going to essentially do the pythagorean theorem.

02:53 You're going to take the square root of the sum of the squares of the vector component.

02:58 So it's going to the square root vx square plus vy squared.

03:03 So that's how you get the value of the, or the magnitude of your vectors.

03:10 So in this case, it be v plus w.

03:11 And lastly, if you're wanting to get the direction, let's say you have a vx, this direction, vy this direction, you'll make this into a right triangle.

03:25 And then if you're wanting to find this angle right here, we know that the tangent of an angle theta is equal to opposite over adjacent.

03:35 So opposite in this case is vy, adjacent is in this case is vx, tangent theta.

03:42 So that means that to find theta, you would take the inverse tangent of v .y over vx.

03:49 Okay, so that's just some review on how to solve, you know, work with vectors.

03:53 So now we're going to go and dive into this problem.

03:56 So we're told that, so again, i'm going to draw out our figure, and we have our vector.

04:00 V here with an angle right here, theta is equal to 50 degrees.

04:07 And we're told that v, it's flying at a speed of 180 miles per hour.

04:15 So that's essentially the magnitude of this angle.

04:17 So we want to split it up into its x and y components.

04:20 So remembering what we did before, the x component of our vector here is going to be equal to our magnitude 180 times the sign of this angle.

04:29 So times sine of 50, degrees.

04:32 That'll give us the component in the x direction.

04:37 So now to get the vy, the component in the y direction, you're going to do the same thing, take your value 180, this time multiplied by the cosine of the angle, so cosine of 50 degrees.

04:48 So that means that our final answer for the vector v is going to be equal to 180 sine 50 degrees, which if you plug that into your calculator, you'll end up with 137 .8 .9, that's in the i direction.

05:09 Then plus your y component will be 180 cosine 50 degrees, which if you plug that into your calculator, you get 115 .70.

05:17 That's in the j direction.

05:19 So that's our v vector.

05:20 And now moving on to our w vector, that's equal to, so we have our w vector in this direction here.

05:31 So it's just in the x direction...

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