Displacement vector points due east and has a magnitude of 2.69 km. Displacement vector points due north and has a magnitude of 7.91 km. Displacement vector points due west and has a magnitude of 8.64 km. Displacement vector points due south and has a magnitude of 4.57 km. Find (a) the magnitude of the resultant vector + + + , and (b) its direction as a positive angle relative to due west.

Add the three displacement vectors. If you do not know how to do this by now, it is time you learned.

Vector has a magnitude of 12.5 units and points due west. Vector points due north.

(a) What is the magnitude of if + has a magnitude of 17.7 units?

To find the magnitude of the resultant vector, we need to add up the individual displacements. Let's break it down step by step:

Step 1: Convert the given magnitudes into x and y components.

For the displacement vector that points due east,
Magnitude = 2.69 km
x-component = 2.69 km (since it points due east, its x-component is positive)
y-component = 0 km (since it doesn't have a north or south component)

For the displacement vector that points due north,
Magnitude = 7.91 km
x-component = 0 km (since it doesn't have an east or west component)
y-component = 7.91 km (since it points due north, its y-component is positive)

For the displacement vector that points due west,
Magnitude = 8.64 km
x-component = -8.64 km (since it points due west, its x-component is negative)
y-component = 0 km (since it doesn't have a north or south component)

For the displacement vector that points due south,
Magnitude = 4.57 km
x-component = 0 km (since it doesn't have an east or west component)
y-component = -4.57 km (since it points due south, its y-component is negative)

Step 2: Add up the x and y components to get the resultant vector components.

x-component of the resultant vector = 2.69 km + (-8.64 km) + 0 km + 0 km = -5.95 km
y-component of the resultant vector = 0 km + 7.91 km + 0 km + (-4.57 km) = 3.34 km

Step 3: Calculate the magnitude of the resultant vector using the Pythagorean theorem.

Magnitude of the resultant vector = √((-5.95 km)^2 + (3.34 km)^2) ≈ 6.80 km

Thus, the magnitude of the resultant vector is approximately 6.80 km (a).

To find the direction of the resultant vector as a positive angle relative to due west, we can use trigonometry.

Step 4: Calculate the angle using the arctan function.

Angle = arctan(y-component of resultant vector / x-component of resultant vector)
= arctan(3.34 km / -5.95 km)

Step 5: Convert the angle from radians to degrees.

Angle (in degrees) = arctan(3.34 km / -5.95 km) * (180 / π) ≈ -30.9° (since it's relative to due west)

Thus, the direction of the resultant vector as a positive angle relative to due west is approximately -30.9° (b).