A spelumker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a dire

A spelumker is surveying a cave. She follows a passage 180 m...

A spelumker is surveying a cave. She follows a passage 180 $\mathrm{m}$ straight west, then 210 $\mathrm{m}$ in a direction $45^{\circ}$ east of south, and then 280 $\mathrm{m}$ at $30^{\circ}$ east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement. (See also Problem 1.73 for a different approach to this problem.)

Key Concepts

Trigonometry in Vector Resolution

Trigonometric principles such as sine and cosine are used to decompose vectors into their horizontal and vertical components when angles are provided. These trigonometric relationships are critical for accurately computing the magnitude and direction of resultant vectors from individual displacement vectors.

Vector Addition

This concept involves combining multiple vectors to determine a resultant vector. When several displacements or forces act on an object, their individual contributions can be summed vectorially. In closed-path problems, the vector sum is zero, meaning that the resultant vector must be equal in magnitude and opposite in direction to the sum of the known vectors.

Component Method

Resolving vectors into their orthogonal components is essential for simplifying the addition process. Each vector is split into horizontal and vertical components using trigonometric functions, which allows for straightforward summation of like components before recombining them to determine the overall magnitude and direction.

Scale Drawings

Scale drawings are graphical representations that maintain proportional relationships between dimensions. In vector problems, they are used to visually illustrate the displacement vectors, allowing one to estimate their combined effect and verify analytical results by measuring distances and angles on the drawing.

Transcript

00:01 For our question, we're asked to draw a diagram of the situation in which this traveler finds herself.

00:08 So she follows a passage 180 meters straight west, or west would be the negative x -axis.

00:14 So that's vector a, okay? and it's straight west, so it's just along the x -axis.

00:19 Then it says that she travels 210 meters in the direction 45 degrees south, 45 degrees east of south.

00:28 And so she travels the vector b next at an angle theta sub b, which is equal to 45 degrees, right? and then we're told that she travels 230 meters, or excuse me, 280 meters, which is vector c at an angle that is 30 degrees east of north.

00:47 So theta c there would be 30 degrees.

00:51 And we're asked to find the magnitude and the direction of her final displacement relative to our initial value, which would be the green line, i call vector d there.

01:00 Okay.

01:01 So in order to do this, we're going to break d into its x and y components, and then we are going to use those to solve for both the magnitude of d and the direction of d.

01:10 It's going to a new page to do this.

01:12 So first, let's consider d of x.

01:14 Well, d of x is just going to be made up of all the x components of the other vectors that were used to make it.

01:19 So it's going to be b of x plus c of x because those both act in the positive x direction minus a of x, which acts in the negative x direction.

01:30 So if you draw right triangles out for b of x and c of x, we find that b of x is equal to the magnitude of b times the cosine of the angle, 45 degrees, plus c of x, which would be c.

01:45 And this one is times the sign of the angle that is made with c.

01:53 And that's the reason it's the sign is because it's opposite over hypotenuse for sign.

01:59 And the opposite there would, the opposite then is going to be c, and the hypotenous then would be c of x, okay? or the, yeah, the opposite would be the c and the hypotenus would be c of x, yeah.

02:15 So we can use c times sine of 30 to get c of x minus, and then a all acts in the x direction.

02:24 So this is just going to be minus the magnitude of a.

02:26 Plugging those values in, for d of x, we find that this is equal to 108, 0 .5 meters...

Please give Ace some feedback