Question

Vector a s is 2.80 cm long and is 60.0° above the x-axis in the first quadrant. vector b s is 1.90 cm long and is 60.0° below the x-axis in the fourth quadrant (fig. e1.35). use components to find the magnitude and direction of (a) a s + b s ; (b) a s − b s ; (c) b s − a s . in each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

Answer 1

Refer to the diagram shown below.

Vector a is 2.8 cm long and 60° above the x-axis in the first quadrant. Therefore it s represented by
$\vec{a} = 2.8 (cos60^{o} \vec{i}+sin60^{o}\vec{j})$

Vector b is 1.9 cm long and 60° below the x-axis in the fourth quadrant. Therefore it is represented by 
$\vec{b}=1.9(cos60^{o}\vec{i}-sin60^{o}\vec{j})$

The vectors may be written as
a = 2.8*(cos60°, sin60°) = (1.445, 2.5028)
b = 1.9*(cos60°, -sin60°) = (0.950, -1.6454)

Part (a)
a + b = (1.445+0.950, 2.5028-1.6454)
         = (2.395, 0.8574)
The magnitude of this vector is
√[2.395² + 0.8574² ] = 2.5438 cm
Its direction is
tan⁻¹ (0.8574/2.395) = 19.7° above the x-axis, in the first quadrant.
The resultant vector is shown in the figure below.

Part (b)
Similarly obtain
a - b = (0.495, 4.1485)
The magnitude is 4.1777.
The direction is 83° above the x-axis, in the 1st quadrant.
The resultant vector is shown in the figure below.

Part (c)
Similarly, obtain
b - a = (-0.495, -4.1483)
The magnitude is 4.1777.
The direction is 83° below the negative x-axis, in the 3rd quadrant.
The resultant vector is shown in the figure below.