Vector A has a magnitude of 23 units and points in the positive y-direction. When vector B is added to A, the resultant vector A+B points in the negative y-direction with a magnitude of 15 units. Find the magnitude of B?
To solve this problem, we can use vector addition and trigonometry.
Let's first analyze the given information:
1. Vector A has a magnitude of 23 units and points in the positive y-direction.
2. When vector B is added to vector A, the resultant vector A+B points in the negative y-direction with a magnitude of 15 units.
To find the magnitude of vector B, we need to consider the net effect on the y-component of the vectors when combined.
Since vector A points in the positive y-direction, its y-component is positive. Let's denote the y-component of vector A as Ay.
Since the resultant vector A+B points in the negative y-direction, its y-component is negative. Let's denote the y-component of vector B as By.
We can express the y-component of the resultant vector A+B as Ay + By = -15.
We know that the magnitude of vector A is 23, so the y-component Ay = 23. Substituting this into the equation, we have:
23 + By = -15.
To isolate By, we can subtract 23 from both sides of the equation:
By = -15 - 23.
By simplifying the equation:
By = -38.
So, the y-component of vector B, By, is -38.
Now, to find the magnitude of vector B, we can use the Pythagorean theorem:
Magnitude of B = sqrt((Bx)^2 + (By)^2).
Since Bx does not affect the y-component of the resultant vector (A+B), we can set it to zero, as it lies in the x-direction.
Magnitude of B = sqrt((0)^2 + (-38)^2).
By simplifying the equation:
Magnitude of B = sqrt(0 + 1444).
Magnitude of B = sqrt(1444).
Magnitude of B = 38.
Therefore, the magnitude of vector B is 38 units.
Let's solve this step by step:
Step 1: Given that vector A has a magnitude of 23 units and points in the positive y-direction.
Step 2: Let's represent vector A in terms of its components - A = (0, 23).
Step 3: We are also given that when vector B is added to A, the resultant vector A+B points in the negative y-direction with a magnitude of 15 units.
Step 4: Let's represent vector B in terms of its components - B = (x, -y), where x represents the x-component and y represents the y-component of vector B.
Step 5: The resultant vector A+B can be written as R = A + B = (0, 23) + (x, -y).
Step 6: Since the resultant vector points in the negative y-direction, the y-component of the resultant vector is -15 units. So, R = (0, -15).
Step 7: Adding the vectors component-wise, we get (0 + x, 23 - y) = (0, -15).
Step 8: Solving for x and y, we have x = 0 and y = 38. Therefore, vector B = (0, 38).
Step 9: Finally, the magnitude of vector B can be calculated using the Pythagorean theorem: |B| = sqrt(x^2 + y^2) = sqrt(0^2 + 38^2) = sqrt(1444) = 38 units.
Therefore, the magnitude of vector B is 38 units.