Find the component form and the magnitude of the vector v. (Express the components of v as fractions. Round the magnitude of v to four decimal places.)
Initial Point: (7/2, -3)
Terminal Point: (1, 2/7)
v=< , >
||v||=
the answers i got are:
v=<1,1>
||v||=(-5/2),(23/7)
but my online assignment says that both of my answers are wrong
v = <(-5/2),(23/7)>
||v|| = √((5/2)^2 + (23/7)^2)
how ever did you get v=<1,1>?
thanks oobleck
To find the component form of vector v, subtract the x and y coordinates of the initial point from the x and y coordinates of the terminal point, respectively.
x-component = (x-coordinate of terminal point) - (x-coordinate of initial point)
= 1 - 7/2
= 2/2 - 7/2
= -5/2
y-component = (y-coordinate of terminal point) - (y-coordinate of initial point)
= 2/7 - (-3)
= 2/7 + 21/7
= 23/7
Therefore, the component form of vector v is v = <-5/2, 23/7>.
To find the magnitude of vector v, use the formula:
||v|| = √((x-component)^2 + (y-component)^2)
||v|| = √((-5/2)^2 + (23/7)^2)
= √(25/4 + 529/49)
= √((625 + 11664)/196)
= √(12289/196)
= √62.6938775510204
≈ 7.9149 (rounded to four decimal places)
Therefore, the magnitude of vector v is ||v|| ≈ 7.9149.
To find the component form of a vector, you need to subtract the corresponding coordinates of the initial and terminal points. Let's calculate it step by step:
Component form of vector v:
vx = x2 - x1
vy = y2 - y1
In this case:
Initial Point: (7/2, -3)
Terminal Point: (1, 2/7)
vx = 1 - 7/2
vy = 2/7 - (-3)
To simplify, let's convert all the fractions into a common denominator of 14:
vx = 2/2 - 7/2
= -5/2
vy = 2/7 + 42/14
= 16/14 + 42/14
= 58/14
= 29/7
Therefore, the component form of vector v is < -5/2, 29/7 >.
To calculate the magnitude of a vector, you can use the formula:
||v|| = √(vx^2 + vy^2)
Applying this formula to the values we have found:
||v|| = √((-5/2)^2 + (29/7)^2)
= √(25/4 + 841/49)
= √(1225/196 + 841/49)
= √(1225/196 + 841*4/196)
= √(1225/196 + 3364/196)
= √(4589/196)
= √(4589) / √(196)
≈ 67.7904 / 14
Rounding the magnitude to four decimal places gives us:
||v|| ≈ 4.8421
Therefore, the correct answers are:
Component form of vector v: < -5/2, 29/7 >
Magnitude of vector v: 4.8421