Vector e is 0.111 m long in a 90.0 direction. vector f is 0.234 m long in a 300 direction. what is the magnitude and direction of their vector sum?

  0.148623∠321.93°

Step-by-step explanation:

You can work these without too much brain work by converting the coordinates to rectangular coordinates, adding those, then converting back to a vector length and angle as may be required.

  0.111∠90° + 0.234∠300° = 0.111(cos(90°), sin(90°)) +0.234(cos(300°), sin(300°))

  = (0, 0.111) + (0.117, -0.2026499) = (0.117, -0.0916499)

The magnitude of this is found using the Pythagorean theorem:

  |E+F| = √(0.117² +(-0.0916499)²)  ≈ 0.148623

The angle can be found using the arctangent function, paying attention to the quadrant. This sum vector has a positive x-coordinate and a negative y-coordinate, so is in the 4th quadrant.

  ∠(E+F) = arctan(y/x) = arctan(-0.0916499/0.117) ≈ -38.07° = 321.93°

The vector sum is E+F = 0.148623∠321.93°.

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You can also draw the triangle that has these vectors nose-to-tail and find the magnitude of the sum using the Law of Cosines. The two sides of the triangle are the lengths of the given vectors and the angle between those can be seen to be 30°. Then the length of the 3rd side of the triangle is ...

  |E+F|² = |E|² +|F|² -2·|E|·|F|·cos(30°) = .012321 +.054756 -.044988 = 0.0220887

  |E+F| = √0.0220887 ≈ 0.148623

The direction of the vector sum can be figured from the direction of vector E and the internal angle of the triangle between vector E and the sum vector. That angle can be found from the law of sines to be ...

  (angle of interest) = arcsin(sin(30°)·|F|/|E+F|) = 128.07°

Then the angle of the vector sum is 450° -128.07° = 321.93°.

A diagram is very helpful for keeping all of the angles straight.

  |E+F| = 0.148623∠321.93°

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^

answer: 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

step-by-step explanation: