Vector A has a magnitude of 71 units and points due west, while vector B has the same magnitude and points due south. Find the magnitude and direction of (a) A + B and (b) A - B . Specify the directions relative to due west.
A force F1of magnitude 5.00 units acts on an object at the origin in a direction è = 45.0° above the positive x-axis. (See the figure below.) A second force F2of magnitude 5.00 units acts on the object in the direction of the positive y-axis. Find graphically the magnitude and direction of the resultant force F1 + F2.
magnitude
1 units
direction
2° counterclockwise from the +x-axis
To find the magnitude and direction of vector A + B, we can use vector addition.
Step 1: Draw vector A due west.
Step 2: Draw vector B due south.
Step 3: Connect the tail of vector B with the tip of vector A. This new vector represents the sum of vector A and vector B, which is vector A + B.
Step 4: Measure the magnitude of vector A + B.
Since vector A and vector B have the same magnitude of 71 units, the magnitude of vector A + B is given by the diagonal of a rectangle with sides 71 units and 71 units. Using the Pythagorean theorem, we can find the magnitude:
Magnitude of A + B = √(71^2 + 71^2) = √(2 * 71^2) = 71√2
Therefore, the magnitude of A + B is 71√2 units.
Step 5: Find the direction of vector A + B relative to due west.
Since vector A points due west and vector B points due south, the direction of vector A + B can be determined by finding the angle between vector A and the diagonal of the rectangle (vector A + B).
This angle can be found using the inverse tangent function:
Angle = atan(71/71) = atan(1)
Since the inverse tangent of 1 is 45 degrees, the direction of vector A + B relative to due west is 45 degrees.
Therefore, the magnitude of A + B is 71√2 units, and the direction of A + B relative to due west is 45 degrees.
To find the magnitude and direction of vector A - B, we can use vector subtraction.
Step 1: Draw vector A due west.
Step 2: Draw vector B due south.
Step 3: Connect the head of vector B with the tail of vector A. This new vector represents the difference of vector A and vector B, which is vector A - B.
Step 4: Measure the magnitude of vector A - B.
Using the same reasoning as in vector addition, since vector A and vector B have the same magnitude of 71 units, the magnitude of vector A - B is given by the side of a square with side length 71 units. Therefore, the magnitude of A - B is 71 units.
Step 5: Find the direction of vector A - B relative to due west.
Since vector A points due west and vector B points due south, the direction of vector A - B can be determined by finding the angle between vector A and the side of the square (vector A - B).
This angle is 90 degrees.
Therefore, the magnitude of A - B is 71 units, and the direction of A - B relative to due west is 90 degrees.
To find the magnitude and direction of vector (a) A + B, we need to add vector A and vector B.
(a) A + B:
- Since vector A points due west, it has a horizontal component equal to -71 units. Its vertical component is zero since it does not have a north or south direction.
- Since vector B points due south, it has a vertical component equal to -71 units. Its horizontal component is zero since it does not have an east or west direction.
- To find the resultant vector, we add the horizontal and vertical components of A and B.
- Horizontal component: -71 + 0 = -71 units
- Vertical component: 0 + (-71) = -71 units
- The magnitude of the resultant vector can be found using the Pythagorean theorem: magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
- magnitude = sqrt((-71)^2 + (-71)^2) = sqrt(2 * 71^2) = sqrt(2) * 71 ≈ 100.28 units.
- The resultant vector (A + B) has a magnitude of approximately 100.28 units.
- The direction of the resultant vector can be found using trigonometry.
- Since the horizontal component is negative (-71), the angle will be in the second quadrant.
- The angle θ can be found using the equation: tan(θ) = vertical component / horizontal component
- tan(θ) = (-71) / (-71) = 1
- θ = arctan(1) ≈ 45 degrees.
- The direction of the resultant vector (A + B) relative to due west is approximately 45 degrees north of west.
To find the magnitude and direction of vector (b) A - B, we need to subtract vector B from vector A.
(b) A - B:
- Since vector A points due west, it has a horizontal component equal to -71 units. Its vertical component is zero.
- Since vector B points due south, it has a vertical component equal to -71 units. Its horizontal component is zero.
- To find the resultant vector, we subtract the horizontal and vertical components of B from A.
- Horizontal component: -71 - 0 = -71 units
- Vertical component: 0 - (-71) = 71 units
- The magnitude of the resultant vector can be found using the Pythagorean theorem: magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
- magnitude = sqrt((-71)^2 + 71^2) = sqrt(2 * 71^2) = sqrt(2) * 71 ≈ 100.28 units.
- The resultant vector (A - B) has a magnitude of approximately 100.28 units.
- The direction of the resultant vector can be found using trigonometry.
- Since the horizontal component is negative (-71), and the vertical component is positive (71), the angle will be in the fourth quadrant.
- The angle θ can be found using the equation: tan(θ) = vertical component / horizontal component
- tan(θ) = 71 / (-71) = -1
- θ = arctan(-1) ≈ -45 degrees.
- The direction of the resultant vector (A - B) relative to due west is approximately 45 degrees south of west.
A = 71[180o]. B = 71[270o].
a. A+B=(71*cos180+71*cos270)+i(71*sin180+71*sin270)
A+B = (-71+0) + i(0-71)
A+B = -71 - i71 = -100.4]45o] or
100.4[45o+180o] = 100.4[225o].
Since the resultant is in Q3, 45o is the
reference angle.
b. A-B=(71*cos180-71*cos270)-i(71sin180-71sin270)
A-B = (-71-0) + i(0-(-71))
A-B = -71 + i71 = -100.4[-45o] or
100.4[-45+180] = 100.4[135o].