Vector vector A has a magnitude of 6.90 units and makes an angle of 46.5° with the positive x-axis. Vector vector B also has a magnitude of 8.00 units and is directed along the negative x-axis. Using graphical methods find the following.
(a) The vector sum vector A + vector B .
Magnitude of A + B______________
Direction of A + B:_______________° counterclockwise from +x-axis
(b) The vector difference vector A - vector B .
Magnitude of A - B: ___________3 units
Direction of A - B: ___________° counterclockwise from +x-axis
23.2
To find the vector sum and difference using graphical methods, you will need to construct a graphical representation of the given vectors and perform the necessary operations.
(a) Vector Sum (A + B):
1. Start by drawing a coordinate system with the x-axis and y-axis.
2. Locate vector A by drawing an arrow with a magnitude of 6.90 units at an angle of 46.5° counterclockwise from the positive x-axis.
3. Next, locate vector B by drawing an arrow with a magnitude of 8.00 units along the negative x-axis.
4. Using the parallelogram or tail-to-head method, draw a vector from the tail of vector B to the head of vector A to represent the vector sum (A + B).
Magnitude of A + B:
To find the magnitude of the vector sum, measure the length of the resultant vector (the vector drawn from step 4) using a ruler or any means of measurement. The measured length represents the magnitude of A + B.
Direction of A + B:
To find the direction of the vector sum, measure the angle between the positive x-axis and the resultant vector (A + B) counterclockwise using a protractor or any means of angular measurement.
(b) Vector Difference (A - B):
1. Use the same coordinate system as before.
2. Locate vector A as described in step 2 of part (a).
3. Locate vector B by drawing an arrow with a magnitude of 8.00 units along the negative x-axis.
4. To find the vector difference (A - B), draw a vector from the tail of vector B to the head of vector A, in the opposite direction.
Magnitude of A - B:
To find the magnitude of the vector difference, measure the length of the resultant vector (the vector drawn from step 4) using a ruler or any means of measurement. The measured length represents the magnitude of A - B.
Direction of A - B:
To find the direction of the vector difference, measure the angle between the positive x-axis and the resultant vector (A - B) counterclockwise using a protractor or any means of angular measurement.
To find the vector sum vector A + vector B, we begin by drawing both vectors on a coordinate plane.
- Vector A has a magnitude of 6.90 units and makes an angle of 46.5° with the positive x-axis.
- Vector B has a magnitude of 8.00 units and is directed along the negative x-axis.
We can start by drawing vector A with its magnitude and angle:
1. Start at the origin (0, 0) of the coordinate plane.
2. Draw a line segment with a length of 6.90 units at an angle of 46.5° counterclockwise from the positive x-axis. Label this line segment as vector A.
Next, we draw vector B:
3. Starting from the endpoint of vector A, draw a line segment with a length of 8.00 units along the negative x-axis. Label this line segment as vector B.
Now, we can find the vector sum by connecting the initial point of vector A to the endpoint of vector B:
4. Draw a line segment connecting the initial point of vector A to the endpoint of vector B. This line segment represents the vector sum, vector A + vector B. Label this line segment as vector A + vector B.
To find the magnitude of vector A + vector B, measure the length of the line segment representing vector A + vector B:
(a) The magnitude of A + B is approximately __________ units.
To find the direction of vector A + vector B, measure the angle counterclockwise from the positive x-axis to the line segment representing vector A + vector B:
(a) The direction of A + B is approximately __________° counterclockwise from the +x-axis.
Now, let's move on to finding the vector difference vector A - vector B:
To find the vector difference, we subtract vector B from vector A. Geometrically, this can be achieved by reversing the direction of vector B:
1. Starting from the initial point of vector A, draw a line segment extending to the opposite direction along the negative x-axis with a length of 8.00 units. Label this line segment as -B (negative B).
To find the vector difference vector A - vector B, connect the endpoint of vector A to the endpoint of -B (negative B):
2. Draw a line segment connecting the endpoint of vector A to the endpoint of -B (negative B). This line segment represents the vector difference, vector A - vector B. Label this line segment as vector A - vector B.
To find the magnitude of vector A - vector B, measure the length of the line segment representing vector A - vector B:
(b) The magnitude of A - B is approximately __________ units.
To find the direction of vector A - vector B, measure the angle counterclockwise from the positive x-axis to the line segment representing vector A - vector B:
(b) The direction of A - B is approximately __________° counterclockwise from the +x-axis.