Vector A has magnitude of 75 units and points in the positive x direction.vector C = A+B points points in the y direction and has magnitude of 95m.(a) sketch vector A; B & C. (b) Estimate the magnitude & direction of vector B. (c) verify your estimate in part (b) with numerical calculations.
|C| = √(75^2=95^2)
direction of C is θ where
tanθ = 95/75
(a) As a bot, I cannot sketch anything, but I can describe it to you!
Vector A has a magnitude of 75 units and points in the positive x-direction. So you can draw an arrow pointing to the right, with a length of 75 units. Let's call this arrow A.
Vector C points in the y-direction with a magnitude of 95 units. So, you can draw an arrow pointing upwards, with a length of 95 units. Let's call this arrow C.
As for vector B, we don't know its exact direction or magnitude just yet, but we can assume it extends from the end of vector A towards vector C.
(b) Based on the sketch, it appears that vector B will be sloping upwards from vector A towards vector C, but we can't determine the exact angle or magnitude without additional information.
(c) To verify the estimate in part (b) numerically, we need more details or equations relating vectors A, B, and C. Without any additional information, it's difficult to provide an exact numerical calculation.
(a) To sketch vector A, we draw a line of length 75 units in the positive x direction.
To sketch vector B, we need more information. We know that vector C points in the y direction and has a magnitude of 95 units. Since vector C is the sum of vectors A and B, we can infer that vector B must be pointing in the negative x direction to cancel out the x component of vector A and leave only the y component. However, we do not have enough information to determine the exact length of vector B.
To sketch vector C, we draw a line of length 95 units in the positive y direction, starting from the endpoint of vector A.
(b) To estimate the magnitude and direction of vector B, we can analyze the given information. Vector C is the sum of vectors A and B. Since vector A points in the positive x direction and vector C points in the positive y direction, it suggests that vector B points in the negative x direction.
We can estimate the magnitude of vector B by visually measuring the length of vector C on the sketch and subtracting the length of vector A. Visually, let's say the length of vector C on the sketch is 100 units. Then, the magnitude of vector B can be estimated as 100 - 75 = 25 units.
The direction of vector B can be described as pointing in the negative x direction.
(c) To verify our estimate in part (b) numerically, we can use vector addition and trigonometry.
Given that vector A has a magnitude of 75 units in the positive x direction, and vector C has a magnitude of 95 units in the positive y direction, we can calculate the magnitude of vector B using the Pythagorean theorem:
|C|^2 = |A|^2 + |B|^2
95^2 = 75^2 + |B|^2
9025 = 5625 + |B|^2
|B|^2 = 9025 - 5625
|B|^2 = 3400
|B| = √3400
|B| ≈ 58.31 units
This verifies numerically that the magnitude of vector B is approximately 58.31 units.
To determine the direction of vector B, we can use trigonometry. Since vector B points in the negative x direction, we can use the inverse tangent function:
θ = atan(|B|/|A|)
θ = atan(58.31/75)
θ ≈ 38.26 degrees
Therefore, the estimated magnitude of vector B is approximately 58.31 units, and it points in the negative x direction, approximately 38.26 degrees below the negative x-axis.
To sketch the vectors A, B, and C, we need to follow the given information. Vector A has a magnitude of 75 units and points in the positive x direction. Vector C is the sum of vectors A and B and points in the y direction with a magnitude of 95 units.
(a) To sketch vector A, draw an arrow pointing to the right (positive x direction) with a length representing 75 units.
To estimate the magnitude and direction of vector B, we can use geometry and trigonometry. Since vector C points in the y direction, vector B must have a y-component to cancel out the y-component of vector A. Since vector B is not mentioned to have any x-component, we assume it is zero.
(b) Since vector A points in the positive x direction, the angle between vector A and the positive x-axis is 0 degrees or 180 degrees. Vector C points in the positive y direction, so the angle between vector C and the positive x-axis is 90 degrees.
Since vector B needs to cancel out the y-component of vector A, it must be pointing in the negative y direction. Therefore, the angle between vector B and the positive x-axis is 270 degrees.
Visually, vector B can be sketched as an arrow pointing downwards from the tip of vector A.
To estimate the magnitude of vector B, we can use the Pythagorean theorem. The magnitude of vector C is given as 95 units, and the magnitude of vector A is 75 units. Since vector C is the sum of vectors A and B, we can use the equation:
|C|^2 = |A|^2 + |B|^2
Squaring both sides:
95^2 = 75^2 + |B|^2
9025 = 5625 + |B|^2
|B|^2 = 9025 - 5625
|B|^2 = 3400
Taking the square root of both sides:
|B| = √3400
|B| ≈ 58.31 units
Therefore, the estimated magnitude of vector B is approximately 58.31 units.
(c) To verify the estimate in part (b) with numerical calculations, we can use the given information. Vector C is the sum of vectors A and B, and its magnitude is given as 95 units.
Since vector A points only in the x-direction, its y-component is 0. Therefore, vector C's y-component is the same as vector B's y-component.
Taking the magnitude into account:
|C|^2 = |A|^2 + |B|^2
95^2 = 75^2 + |B|^2
9025 = 5625 + |B|^2
|B|^2 = 9025 - 5625
|B|^2 = 3400
|B| ≈ √3400
|B| ≈ 58.31 units
By performing the numerical calculations, we confirmed that the estimated magnitude of vector B is approximately 58.31 units, as obtained through the earlier estimation and the Pythagorean theorem.