(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a directio

(a) Repeat the problem two problems prior, but for the...

Key Concepts

Vector Addition

Vector addition involves summing vectors by combining their individual components along the coordinate axes. This concept is essential in determining the net displacement or resultant vector when multiple vector quantities, each with their magnitude and direction, are involved. By breaking each vector into its x (horizontal) and y (vertical) components, the overall effect of sequential movements or forces can be computed by summing these components respectively.

Vector Subtraction

Vector subtraction is used to find the difference between two vectors, which can represent, for example, the change in position or the reversal of a displacement. It involves subtracting the corresponding components of the vectors to compute a new vector that points from the tip of the subtracted vector to the tip of the minuend vector. This operation is closely related and often visualized as adding a vector that is equal in magnitude but opposite in direction.

Component Resolution

Resolving vectors into components is a fundamental technique in vector analysis that breaks a vector into perpendicular elements, typically along the x and y axes. This is achieved using trigonometric functions such as cosine and sine for the horizontal and vertical components, respectively. Component resolution allows complex vector problems to be simplified into independent one-dimensional problems.

Direction Conventions and Angle Measurement

Understanding and correctly interpreting directional terms, such as 'east of south' or angles measured from a reference axis, is critical in vector problems. This includes knowing how to convert directional descriptions into standard angles used in trigonometric calculations, and recognizing that a change in interpretation of the direction can lead to a sign change in the vector components. Aligning the vector's direction with the coordinate system ensures that subsequent operations like addition or subtraction are accurate.

Transcript

00:01 Vector's vector a, vector b, we know that vector a is going to be 12 .0 meters.

00:05 This is going to be 20 degrees west of north.

00:13 And then we have vector b 20 .0 meters.

00:16 Now this is going to be 40 degrees north of east.

00:23 And we're trying to find r, which is essentially equaling a plus b.

00:32 So we can then say that here, a sub x, a sub, y b sub x b sub y a sub x is going to be equaling 12 .0 meters cosine now here is the difficulty here we have 20 degrees west of north we're in the second quadrant so this is going to be negative 12 .0 meters cosine of 20 degrees west of north is the same as saying actually 70 degrees north of west so negative 12 .0 meters cosine of of 70 degrees, this is equaling negative 4 .104 meters.

01:20 And then we have a sub y, 12 .0 meters, sign of 70 degrees.

01:25 We're in the second quadrant, so we know that this is gonna be positive.

01:28 11 .27 meters.

01:30 B sub x, now 40 degrees north of east, we're in the first quadrant.

01:35 So we're gonna have a positive x and a positive y.

01:37 So this would be 20 .0 meters, a cosine of 40 degrees, 20 .0.

01:44 Meters sign of 40 degrees for the x component we have 15 .32 meters for the y component we have 12 .85 meters and finding r sub x this is simple we just add these two together so it would be 15 .32 minus 4 .104 and r sub x is equaling 11 rather positive 11 .21 meters my apologies our sub y is equaling again you just add these up 11 .27 plus 12 .85 and we have our sub y equaling 24 .12 meters now at this point we need to find the magnitude of the result in vector so this would be equaling essentially a 11 .21 meters quantity squared plus 24 .12 meters quantity squared, all raised to the one half power.

02:59 So the sum of the squares is raised to the one half power.

03:02 This is equaling 26 .59 meters.

03:05 This would be our magnitude for our result in vector.

03:09 Now, our direction would be equalling arc 10 of the y component divided by the x component.

03:15 And in this case, we have 24 .12 meters divided by 11 .21 meters.

03:24 Now, this is equaling 65 .07 degrees.

03:29 But going back, we know that here we have positive x component and a positive y component for the resultant vector, which actually means that here we are going to be in the first quadrant.

03:41 So this would be 65 .07 degrees north of east.

03:46 Now, at this point, we can actually, this would be our final answer for part a, and now we can go to part b, where here we have the exact same.

03:57 However, now, vector a, vector b, vector a is going to be equaling 20 .0 meters, and this would be 40 degrees south of west.

04:18 And now we have 12 .0 meters, 20 .5 .2...

Please give Ace some feedback