Finding Vector v. ASAP!!
- Thread starter Jac
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mmm4444bot
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Unless you have some additional information that relates vectors u and v, I don't see any way to find the components of vector v.
What you've told us so far amounts to me asking you the following.
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I placed a broom on my front lawn so that the handle is pointing toward NE.
I placed a second broom on my front lawn; its handle is five feet long.
What direction is the handle on the second broom pointing?
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Can you understand why there is really no way for you to answer this question? The reasons why are the same reasons why I cannot answer your question. There is not enough information.
If you're working on an exercise assigned by an instructor, then please post the entire exercise.
mmm4444bot
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The missing piece of information in your original post is that vector v has the same direction as vector u.
In other words, v is the same as u, except that v is 5 units long, whereas u is shorter.
Do you know?
If the tail of u is placed at the origin on an xy-coordinate system, then u lies on the line y = x.
In other words, u makes a 45-degree angle with the positive x-axis, so the head of u lies in Quadrant I at (3,3).
Since v has the same direction, it will also lie on the line y = x, if we position it on the coordinate system with its tail also at the origin.
Therefore, the components of v will be the same as the (x, y) coordinates of the point in Quadrant I that is five units from the origin.
Do you know how to find the coordinates of the point in Quadrant I that lies on the line y = x and is located five units away from the origin? (There are different methods.)
pka
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The length of u is \(\displaystyle \left\| u \right\| = \sqrt {3^2 + 3^2 } = 3\sqrt 2\).Jac said:
Now multiply u by five times its own multiplicative inverse:\(\displaystyle v=\left\langle {\frac{{15}}{{3\sqrt 2 }},\frac{{15}}{{3\sqrt 2 }}} \right\rangle = \left\langle {\frac{{5\sqrt 2 }}{2},\frac{{5\sqrt 2 }}{2}} \right\rangle\).
There you have it.
mmm4444bot
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I'll show you the steps for the original exercise. You can use the same strategy.
u = < 3, 3 >
||v|| = 5
Find the componenets of v
One strategy is to first calculate the components for a unit vector (i.e., a vector whose length is one unit) in the same direction as vector u, and then multiply each of those components by 5 because that increases the vector's length to five units.
Finding the unit-vector components takes two steps; increasing the unit vector to five units in length is the third step.
STEP 1: Calculate the magnitude of u
||u|| = sqrt(3^2 + 3^2)
= sqrt(9 + 9)
= sqrt(18)
= 3 sqrt(2)
STEP 2: Calculate the unit components of u
u* = < 3/[3 sqrt(2)], 3/[3 sqrt(2)] >
= < 1/sqrt(2), 1/sqrt(2) >
= < sqrt(2)/2, sqrt(2)/2 >
This unit vector for u has magnitude 1.
To get a vector with magnitude 5, in the same direction, we multiply the components by a factor of 5.
STEP 3: Increase both components by a factor of 5
v = < 5 sqrt(2)/2, 5 sqrt(2)/2 >
If you need more help, then please reply with your specific questions. Simply stating that you don't understand tells us nothing about why you're stuck.
If I wrote anything that you do not understand, then you need to tell us specifically what it is that you do not understand because vague requests for help are nearly meaningless.
Perhaps, you don't understand how to find the length of a vector. Perhaps, you've forgotten how to do arithmetic with radicals. Perhaps, you have not yet learned what a vector is. I dunno what you dunno, until you tell me. Okay?
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babyj1970 said:
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I'll show you the steps for the original exercise. You can use the same strategy.
u = < 3, 3 >
||v|| = 5
Find the componenets of v
One strategy is to first calculate the components for a unit vector (i.e., a vector whose length is one unit) in the same direction as vector u, and then multiply each of those components by 5 because that increases the vector's length to five units.
Finding the unit-vector components takes two steps; increasing the unit vector to five units in length is the third step.
STEP 1: Calculate the magnitude of u
||u|| = sqrt(3^2 + 3^2)
= sqrt(9 + 9)
= sqrt(18)
= 3 sqrt(2)
STEP 2: Calculate the unit components of u
u* = < 3/[3 sqrt(2)], 3/[3 sqrt(2)] >
= < 1/sqrt(2), 1/sqrt(2) >
= < sqrt(2)/2, sqrt(2)/2 >
This unit vector for u has magnitude 1.
To get a vector with magnitude 5, in the same direction, we multiply the components by a factor of 5.
STEP 3: Increase both components by a factor of 5
v = < 5 sqrt(2)/2, 5 sqrt(2)/2 >
If you need more help, then please reply with your specific questions. Simply stating that you don't understand tells us nothing about why you're stuck.
If I wrote anything that you do not understand, then you need to tell us specifically what it is that you do not understand because vague requests for help are nearly meaningless.
Perhaps, you don't understand how to find the length of a vector. Perhaps, you've forgotten how to do arithmetic with radicals. Perhaps, you have not yet learned what a vector is. I dunno what you dunno, until you tell me. Okay?
?