A dog searching for a bone walks 4.31 m south, then runs 8.93 m at an angle 22.0 north of east, and finally walks 10.0 m west. Find the magnitude of the dog's resultant displacement vector.

Add the three vectors.

The +y (north) component of the sum is
X = -4.31 +8.93 sin 22

The +x (east) component of the sum is
Y = 8.93 cos 22 - 10.0

The magnitude of the displacement vector is sqrt(X^2 + Y^2)

The direction (counterclockwise from east) is arctan Y/X

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To find the magnitude of the dog's resultant displacement vector, we need to determine the total distance and direction it has traveled.

Let's represent the south direction as negative (-) and the east and west directions as positive (+).

The dog walks 4.31 m south, which means it has traveled -4.31 m.

Next, the dog runs 8.93 m at an angle 22.0° north of east. To find the eastward component, we can calculate the cosine of the angle multiplied by the distance:

eastward component = cos(22.0°) * 8.93 m

Similarly, to find the northward component, we can calculate the sine of the angle multiplied by the distance:

northward component = sin(22.0°) * 8.93 m

Since the angle is north of east, the northward component is positive.

The dog walks 10.0 m west, which means it has traveled -10.0 m.

Now, we can calculate the total eastward displacement:

eastward displacement = 8.93 m * cos(22.0°)

And the total northward displacement:

northward displacement = 8.93 m * sin(22.0°)

To find the total eastward displacement, we can add up the eastward and westward components:

total eastward displacement = eastward displacement + (-10.0 m)

To find the total northward displacement, we can add up the northward and southward components:

total northward displacement = northward displacement + (-4.31 m)

Finally, we can find the magnitude of the resultant displacement vector (R) using the Pythagorean theorem:

R = sqrt((total eastward displacement)^2 + (total northward displacement)^2)

By plugging the values into the equation and performing the necessary calculations, we can find the magnitude of the dog's resultant displacement vector.