1)Vector A has a magnitude of 26 and is pointing at an angle a=27° east of north

Question

1)Vector A has a magnitude of 26 and is pointing at an angle a=27° east of north. Vector B has a magnitude of 21 and is pointing at an angle b=13° south of west. Find the magnitude and direction of A+B (the sum of two vectors is more commonly known as the resultant vector).

a)What is magnitude of the resultant vector?

b)What direction is the resultant vector pointing in which of the following?

It has no magnitude so the direction is irrelevant.
Due east (along the positive x-axis)
Due north (along the positive y-axis)
Due west (along the negative x-axis)
Due south (along the negative y-axis)
A northeasterly direction (first quadrant)
A northwesterly direction (second quadrant)
A southwesterly direction (third quadrant)
A southeasterly direction (fourth quadrant)

c)What is the angle between the resultant vector and the horizontal? If the magnitude of the resultant is zero, then enter 'none'.

Explanation / Answer

This is easiest if we break up the vectors into x and y components:

x = r cos()

y = r sin()

The tricky part is that the we have to reconize what the angles are.

27 degrees east of north is 90-27 = 63 degrees

13 degrees south of west is 180+13 = 193 degrees

Now that we have that, we can find our values for the resultant vector:

x´= 26 cos(63) + 21 cos(193) = -8.658

y´= 26 sin(63) + 21 sin(193) = 18.442

We now just use some trigonometry to find the resultant magnitude and direction:

r´=(x´2 + y´2) = (-8.658)2 + 18.4422) = 20.373

= tan-1(y/x) = tan-1(18.442/-8.658) = -64.85 degrees (i.e. in quadrant 4)

We can now answer the questions with ease:

a) 20.373

b) A southeasterly direction (fourth quadrant)

c) -64.85 degrees = 360 - 64.85 = 295.15 degrees

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