Vector
A
has a magnitude of 29 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 and direction of
B?
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vector A has a magnitude of 29 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 14 units. find the magnitude and direction of B
To find the magnitude and direction of vector B, we can use the given information about vector A and the resultant vector A + B.
1. Magnitude of A: The magnitude of vector A is given as 29 units.
2. Magnitude of A + B: The magnitude of the resultant vector A + B is given as 15 units.
3. Direction of A: Vector A points in the positive y-direction.
Now, let's calculate the magnitude and direction of vector B step by step:
Step 1: Magnitude of B
Since vector B is added to vector A to get the resultant vector A + B, we can use the Pythagorean theorem to find the magnitude of vector B.
Magnitude of A + B = sqrt((Magnitude of A)^2 + (Magnitude of B)^2)
15 = sqrt(29^2 + (Magnitude of B)^2)
225 = 841 + (Magnitude of B)^2
(Magnitude of B)^2 = 225 - 841
(Magnitude of B)^2 = -616
Since the magnitude of a vector cannot be negative, it means there is no real magnitude for vector B that satisfies the given conditions. Therefore, vector B cannot be determined.
Step 2: Direction of B
Without the magnitude of vector B, we cannot determine its direction.
Therefore, the magnitude and direction of vector B cannot be determined based on the given information.
To find the magnitude and direction of vector B, we can use the given information about the resultant vector A + B.
1. Magnitude of Vector A = 29 units
2. Magnitude of Resultant Vector A + B = 15 units
Since vector A points in the positive y-direction, we can assume it is a vertical vector.
Now, let's analyze the given information. When vector B is added to vector A, the resultant vector A + B points in the negative y-direction. This means that vector B is pointing in the opposite direction from vector A.
Since vector A has a magnitude of 29 units, and the magnitude of the resultant vector A + B is 15 units, we can conclude that the magnitude of vector B is:
Magnitude of B = Magnitude of A + B - Magnitude of A
= 15 units - 29 units
= -14 units
The negative sign indicates that vector B is directed in the opposite direction of vector A.
Therefore, the magnitude of vector B is 14 units, and it points in the negative y-direction.