1. A vector is defined as having magnitude of 15 m and a direction of East. Multiply this vector by the scalar value of –6. What is the resultant vector’s magnitude and direction?
A. 90 m East
B. 90 m West
C. -90 m East
D. -90 m West
2. What are the components of a vector C→ if its magnitude is 8.9 m/s and it makes an angle of –40° with the +x-axis?
A. Cx = –5.72 m/s, Cy = 6.82 m/s
B. Cx = 6.82 m/s, Cy = –5.72 m/s
C. Cx = –0.643 m/s, Cy = 0.766 m/s
D. Cx = 0.766 m/s, Cy = –0.643 m/s
Help!!!
#1 multiplying by a scalar does not change the direction
Just multiply each component by the scalar value
#2 as always, apply the usual conversion from rectangular to polar form:
x = r cosθ
y = r sinθ
In this case, r is just the length of the vector.
"A vector is defined as having magnitude of 15 m and a direction of East."
-----> [ 15, 0]
so -6[15,0] = [-90,0)]
how far is that point from the origin? 90
in which direction does it lie? west
"magnitude is 8.9 m/s and it makes an angle of –40° with the +x-axis"
----> [8.9cos(-40) , 8.9sin(-40) ] =
Thank you I got it now.
To solve these problems, we need to understand vector operations and how to calculate vector components.
1. To find the resultant vector's magnitude and direction, we first need to multiply the original vector by the scalar value of -6. Remember, a scalar only affects the magnitude of the vector, not its direction. Multiplying a vector by a negative scalar will change the direction but keep the magnitude the same.
Given vector magnitude: |V| = 15 m
Given vector direction: East
Multiplying the vector magnitude by the scalar value of -6:
|-6V| = |-6| * |V| = 6 * 15 = 90 m
Since the scalar value is negative, the direction will change. East means moving towards the right in the positive direction. To move towards the left or the West direction, we need to reverse the original direction.
Resultant vector's magnitude: 90 m
Resultant vector's direction: West
Therefore, the answer is:
D. -90 m West
2. To find the components of a vector, we can use the trigonometric relationships between the vector magnitude, angle, and the components along the x and y axes.
Given vector magnitude: |C| = 8.9 m/s
Given angle: -40° with the +x-axis
To find the x-component (Cx) and y-component (Cy), we use the following relationships:
Cx = |C| * cos(theta)
Cy = |C| * sin(theta)
Substituting the given values:
Cx = 8.9 m/s * cos(-40°)
Cy = 8.9 m/s * sin(-40°)
Using a calculator:
Cx = 8.9 m/s * 0.766 = 6.82 m/s
Cy = 8.9 m/s * (-0.643) = -5.72 m/s
Therefore, the answer is:
B. Cx = 6.82 m/s, Cy = -5.72 m/s
I hope this explanation helps! Let me know if you have any further questions.