Displacement vector A points due east and has a magnitude of 2.26 km. Displacement vector B points due north and has a magnitude of 2.62 km. Displacement vector C points due west and has a magnitude of 2.6 km. Displacement vector D points due south and has a magnitude of 2.6 km. Find the magnitude and direction (relative to due east) of the resultant vector A + B + C + D.
Just add the vectors:
(2.26,0)
(0,2.62)
(-2.6,0)
(0,-2.6)
---------
(-0.34,-0.02) = 0.34 at W3.37°S
East (along x axis)
2.26 -2.60
so
-.34
North (along y axis)
2.62 -2.60
so
+.02
magnitude = sqrt (.34^2+.02^2)
= .341
direction (quadrant 2)
sin angle to x axis = .02/.341
so angle above negative x axis = 3.36 deg above -y axis
angle to + x axis = 180 -3.36 = 177 deg counterclockwise
angle to x axis clockwise (like a compass) = 180+3.36 = 183.3
By the way I bet you have at least one typo, but the method should work.
Thanks for checking
grumble grumble
To find the magnitude and direction of the resultant vector A + B + C + D, we first need to add the vectors together.
Adding Vectors:
To add vectors, we need to break them down into their horizontal (x-component) and vertical (y-component) parts.
For vector A:
- It points due east, so its x-component is positive (+2.26 km) and its y-component is zero (0 km).
For vector B:
- It points due north, so its x-component is zero (0 km) and its y-component is positive (+2.62 km).
For vector C:
- It points due west, so its x-component is negative (-2.6 km) and its y-component is zero (0 km).
For vector D:
- It points due south, so its x-component is zero (0 km) and its y-component is negative (-2.6 km).
Now, we can add the x-components and y-components separately.
Adding the x-components:
2.26 km - 2.6 km = -0.34 km
Adding the y-components:
2.62 km - 2.6 km = 0.02 km
The resultant vector has an x-component of -0.34 km and a y-component of 0.02 km.
Magnitude and Direction:
To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude = sqrt((-0.34 km)^2 + (0.02 km)^2)
Magnitude ≈ sqrt(0.1156 km^2 + 0.0004 km^2)
Magnitude ≈ sqrt(0.116 km^2)
Magnitude ≈ 0.34 km
So, the magnitude of the resultant vector A + B + C + D is approximately 0.34 km.
To find the direction of the resultant vector relative to due east, we can use the trigonometric function tangent:
Direction = atan(y-component / x-component) + 180 degrees
Direction = atan(0.02 km / -0.34 km) + 180 degrees
Direction ≈ 175.9 degrees (rounded to one decimal place)
Therefore, the magnitude of the resultant vector A + B + C + D is approximately 0.34 km, and its direction relative to due east is approximately 175.9 degrees.