Suppose you first walk 12.0 m in a direction 20^∘ west of north and then 20.0 m in a direction

Suppose you first walk 12.0 m in a direction 20^∘ west of...

Key Concepts

Trigonometric Relationships in Vectors

Trigonometric functions such as sine and cosine are used to relate the angles provided by directional instructions to the horizontal and vertical components of vectors. Understanding these relationships is crucial for accurately decomposing vectors and determining the final location and its orientation relative to a reference direction.

Vector Addition

Vector addition is the process of combining two or more vectors to find a resultant vector, which represents the cumulative effect of the individual displacements. By adding corresponding components of each vector, one can determine both the overall magnitude and the exact direction from the starting point.

Resultant Magnitude and Direction

After calculating the sum of individual vector components, the magnitude of the resultant vector is determined using the Pythagorean theorem, and its direction is found using inverse trigonometric functions. This process yields a clear measure of how far an object is displaced and the compass direction of the net movement.

Displacement Vectors

Displacement vectors represent the change in position of an object, indicating both magnitude and direction relative to a starting point. They are used in problems involving motion to succinctly capture both how far and in which direction an object has moved, regardless of the path taken.

Vector Decomposition

Vector decomposition involves breaking a vector into its horizontal and vertical components using trigonometric functions. This is an essential technique that translates directional descriptions (such as those given in compass bearings) into components that can be easily added using standard algebraic methods.

Transcript

00:02 In this example, we want to figure out how far away from our starting position we end up, given that we first travel 12 meters in a direction 20 degrees west of north, and then immediately after that we travel 20 meters in a direction 40 degrees south of west.

00:17 So we want to figure out how far away from our starting position we ended up, and also what the direction of that resultant vector is.

00:27 Okay.

00:29 So first, let's just draw out what these paths.

00:33 You're going to look like.

00:37 So here's our compass here, north -south, east and west.

00:46 And the first part says that we travel 12 meters in a direction 20 degrees west of north.

00:53 So we're going to start at the north axis and then we're going to draw out an angle 20 degrees towards the west axis from that.

01:04 So maybe something like this.

01:10 So this would represent our first part of our walk.

01:14 And we'll give this vector a name a.

01:18 And remember a is going to have a magnitude of 12.

01:22 And it makes an angle 20 degrees right here.

01:28 Okay.

01:28 Now for the second part of our walk, we're going 20 meters in the direction 40 degrees south of west.

01:37 So if we drew that at the origin, let's see, we would start at the west.

01:44 Axis here, and then we would go 40 degrees towards the south axis.

01:50 So something like that.

01:53 So that's what the vector should look like if it started at the origin.

01:57 But of course, our vector is going to start at the position we ended up at after our first part of our path.

02:06 So it should be something like this.

02:09 And we know it's going to be a bit longer than a because this one is 20 meters instead of 12.

02:16 So maybe this is what our second.

02:17 One looks like here.

02:23 So for this guy, we will give it a name of b.

02:27 And again, we know what the magnitude of b is.

02:29 It's just 20 meters.

02:31 And we also know that the angle with the horizontal here is going to be 40 degrees.

02:42 Okay.

02:43 So now, what we want to do, right, is calculate the magnitude and direction of the resultant vector.

02:51 Remember, the resultant is just going to point from where we started to where we ended up.

02:57 So we'll call this vector r.

03:00 And actually, we'll even give it a different color just so it's clear.

03:06 So our result in vector should look like that.

03:11 And in order to calculate our result in vector, we know that it's just going to be the vector sum of a and b, the individual pieces of our path, right? and since this is a vector sum, we know that we can calculate r just by summing the components of a and b together.

03:30 So at this point we'll want to find all of the components of a and b so that we can get r.

03:37 So we'll want a sub x, a sub y, b sub x and b sub y.

03:51 Okay.

03:52 So starting with a sub x, from our diagram here, we can see that the sign function will help us get the x component of vector a.

04:07 So if we drew vector a and then made a triangle with vector a, just by using our axes like this...

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