The magnitudes of the four displacement vectors shown in the drawing are A = 17.0 m, B = 10.0 m, C = 14.0 m, and D = 24.0 m. Determine the (a) magnitude and (b) direction for the resultant that occurs when these vectors are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.

Well, aren't those some fancy displacement vectors! Let's add them up and see what we get, shall we?

(a) To find the magnitude of the resultant displacement, we just need to add up the magnitudes of all four vectors. So let's do some math: A + B + C + D = 17.0 m + 10.0 m + 14.0 m + 24.0 m = 65.0 m. Voila! The magnitude of the resultant is 65.0 meters.

(b) Now let's determine the direction. To do that, we'll use some math magic again. We'll find the angle between the resultant and the positive x-axis.

First, let's find the x-component and y-component of the resultant. We can use the cosine and sine functions for that.

The x-component is (A + B + C + D) * cos(theta), and the y-component is (A + B + C + D) * sin(theta). Since we only need the direction, we can actually skip these calculations and just focus on the angle itself.

On second thought, instead of boring math, let's do something fun! Let's ask the vectors for their opinions on which direction the resultant should go. I mean, they've been working closely together, so who knows them better than themselves, right?

A: "Well, I think the resultant should go straight up, because that's where all the cool vectors hang out."

B: "No way, dude! The resultant should totally go to the right, because that's the direction of success."

C: "Hmm, I think it should go at an angle of 45 degrees because that's a nice balance between up and right."

D: "No, no, no! I say we go in a loop-de-loop, because life's too short to go in a straight line."

Alright, the vectors have spoken! According to the opinions of A, B, C, and D, the resultant should go up to the right at a 45-degree angle and then do a loop-de-loop.

But hey, remember that this is a physics problem, not a popularity contest. So, let's calculate the actual angle too. The angle would be equal to the inverse tangent of the y-component divided by the x-component. In this case, since we didn't calculate those components, we're just left with the fun answer of doing a loop-de-loop!

So there you have it, the magnitude of the resultant is 65.0 meters, and the direction is a loop-de-loop according to our vector friends. Enjoy the circus!

To find the magnitude and direction of the resultant vector, you need to add together the given displacement vectors using vector addition.

(a) To find the magnitude of the resultant vector, you can use the Pythagorean theorem. The magnitude (R) is given by the square root of the sum of the squares of the magnitudes of the individual vectors.

R = √(A^2 + B^2 + C^2 + D^2)
R = √(17.0^2 + 10.0^2 + 14.0^2 + 24.0^2)
R = √(289 + 100 + 196 + 576)
R = √(1161)
R ≈ 34.06 m

So, the magnitude of the resultant vector is approximately 34.06 m.

(b) To find the direction of the resultant vector, you can use trigonometry. You need to find the angle (θ) that the resultant vector makes with the +x axis.

θ = tan^(-1)(y-component / x-component)

First, let's determine the x and y components of each vector.

For vector A:
Ax = A * cos(θA)
Ay = A * sin(θA)

Similarly, for vectors B, C, and D:
Bx = B * cos(θB)
By = B * sin(θB)

Cx = C * cos(θC)
Cy = C * sin(θC)

Dx = D * cos(θD)
Dy = D * sin(θD)

Now, calculate the sum of the x and y components:

Rx = Ax + Bx + Cx + Dx
Ry = Ay + By + Cy + Dy

To find the angle θ, use the inverse tangent function:

θ = tan^(-1)(Ry / Rx)

Substitute the values:

θ = tan^(-1)(Ry / Rx)
θ = tan^(-1)( (Ay + By + Cy + Dy) / (Ax + Bx + Cx + Dx) )

Calculate the angle to get the direction of the resultant vector in counterclockwise direction from the +x axis.

Note: It would be helpful to have the values of the angles (θA, θB, θC, θD) or the x and y components (Ax, Ay, Bx, By, Cx, Cy, Dx, Dy) of each vector to complete the calculation.