three displacement vectors of a croquet ball are shown in figure p3.49, where |a| = 20.0 units, |b| = 40.0 units, and |c| = 30.0 units. find (a) the resultant in unit–vector notation and (b) the magnitude and direction of the resultant displacement
Need more INFO on direction of vectors.
To find the resultant displacement, we need to find the sum of the three displacement vectors.
(a) Unit-Vector Notation:
Let's label the three displacement vectors as follows:
Vector a = 20.0 units
Vector b = 40.0 units
Vector c = 30.0 units
To find the resultant in unit-vector notation, we need to add the vectors component-wise. We get:
Resultant = a + b + c
In unit-vector notation, the resultant would be represented as R = Rxi + Ryj + Rzk, where Rx, Ry, and Rz are the components of the resultant vector.
Now, let's calculate each component separately:
Rx = ax + bx + cx
= 20.0 units + 40.0 units + 30.0 units
= 90.0 units
Ry = ay + by + cy
= 0 units + 0 units + 0 units
= 0 units
Rz = az + bz + cz
= 0 units + 0 units + 0 units
= 0 units
Therefore, the resultant in unit-vector notation is:
R = 90.0xi + 0j + 0k
(b) Magnitude and Direction:
To find the magnitude of the resultant displacement, we use the Pythagorean theorem:
Magnitude (R) = sqrt((Rx)^2 + (Ry)^2 + (Rz)^2)
= sqrt((90.0 units)^2 + (0 units)^2 + (0 units)^2)
= sqrt(8100 units^2)
= 90.0 units
To find the direction of the resultant, we can calculate the angle it makes with the x-axis using the arccos function:
Direction (θ) = arccos(Rx / R)
= arccos(90.0 units / 90.0 units)
= arccos(1)
= 0 radians
Therefore, the magnitude of the resultant is 90.0 units, and the direction is 0 radians.
To find the resultant displacement of the croquet ball, we need to add the three displacement vectors together.
(a) To express the resultant in unit-vector notation, we need to find the x, y, and z-components of the resultant vector. Let's label the three vectors as follows:
Vector a: A
Vector b: B
Vector c: C
Now, let's break down each vector into its x, y, and z-components. Assume the x-component is represented by i, the y-component by j, and the z-component by k.
For vector A:
A = 20*i + 0*j + 0*k
For vector B:
B = 0*i + 40*j + 0*k
For vector C:
C = 0*i + 0*j + 30*k
Now, we can add these vectors together to find the resultant vector:
Resultant vector, R = A + B + C
R = (20*i + 0*j + 0*k) + (0*i + 40*j + 0*k) + (0*i + 0*j + 30*k)
R = 20*i + 40*j + 30*k
So, the resultant displacement in unit-vector notation is R = 20*i + 40*j + 30*k.
(b) To find the magnitude and direction of the resultant displacement, we can use the Pythagorean theorem and trigonometry. The magnitude of the resultant is given by:
Magnitude, |R| = sqrt((20^2) + (40^2) + (30^2))
|R| = sqrt(400 + 1600 + 900)
|R| = sqrt(2900)
|R| ≈ 53.8 units
To find the direction of the resultant, we can use inverse trigonometric functions. We can find the angle between the resultant vector and the x-axis:
θ = arctan(Ry / Rx)
θ = arctan(40 / 20)
θ = arctan(2)
θ ≈ 63.4 degrees
Therefore, the magnitude of the resultant displacement is approximately 53.8 units, and the direction is approximately 63.4 degrees with respect to the positive x-axis.