Question about introductory vectors.

In summary, the problem asks to find a vector with the same direction as u, given a magnitude of 6 and u = < 0,3 >. Using the equation u = v/||v|| and substituting the numbers, it can be solved for v to get v = < 0,6 >. However, the direction of u is < 0,1 >, so to find a vector with the same direction and magnitude of 6, the unit vector in the direction of u must be multiplied by 6, resulting in < 0,6 > as the components of the new vector. The unit vector in the direction of u is found by dividing u by its magnitude, giving < 0,1
  • #1
icesalmon
270
13

Homework Statement



Given ||v|| = 6 and u = < 0,3 >
find the vector v with the given magnitude and the same direction as u


Homework Equations



u = v / || v ||

The Attempt at a Solution



It seems simple enough, substitute the appropriate numbers into the equation and solve for v so < 0,3 > = v / 6 so v = 6 * < 0,3 > or v = < 0,18 >. The answer is that v = < 0,6 >. I'm really not sure what is going on.
 
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  • #2
Answer these questions: if u = <0, 3>, what is its direction?
If a vector of magnitude 6 is in the same direction as vector u, then what are the components of this new vector?
 
  • #3
if u = < 0,3 > it's direction, magnitude, or size is 3
< 0,6 > are the components of a vector with magnitude 6. Oh, and because it's direction is the same all you have to change is it's magnitude, okay I understand.

That doesn't answer why I didn't get v = < 0,6 > using the above equation.

What you're saying makes sense, but I don't know if it's that easy with the following problems.
 
  • #4
icesalmon said:
if u = < 0,3 > it's direction, magnitude, or size is 3
Its magnitude or size is 3, but its direction is <0, 1>. To get this, multiply the vector by the reciprocal of the magnitude. The direction of a vector v is the unit vector (1/|v|) v.
icesalmon said:
< 0,6 > are the components of a vector with magnitude 6. Oh, and because it's direction is the same all you have to change is it's magnitude, okay I understand.

That doesn't answer why I didn't get v = < 0,6 > using the above equation.

What you're saying makes sense, but I don't know if it's that easy with the following problems.
 
  • #5
Okay, so I need to re-read this section before I put a pencil to paper. I clearly don't have any sort of grasp for these things. Thanks for your help, i'll post back if I have any further questions.
 
  • #6
To find a vector of magnitude "x" in the direction of vector [itex]\vec{v}[/itex], you have to multiply a unit vector in that direction by x. A unit vector in the same direction as v is given by
[tex]\frac{\vec{v}}{|\vec{v}|}[/tex]

In this particular problem, [itex]\vec{v}= <0, 3>[/itex], which has length 3, so a unit vector in that direction is
[tex]\frac{<0, 3>}{3}= <0, 1>[/tex]

A vector of length 6 in the direction of [itex]\vec{v}\itex] is
[tex]6\frac{<0, 3>}{3}= 6<0, 1>= <0, 6>[/tex]
 
  • #7
okay, I understand the problem now. A lot more than I did before, it's not difficult, and maybe I should have posted it in the pre-calculus section. But this is where we started off in my Calculus II course. Thanks HallsofIvy.
 

Related to Question about introductory vectors.

1. What are vectors?

Vectors are mathematical objects that have both magnitude (size) and direction. They are often used to represent physical quantities, such as velocity or force.

2. How are vectors represented?

Vectors can be represented graphically with an arrow pointing in the direction of the vector and a length proportional to its magnitude. They can also be represented algebraically using coordinates or components.

3. How are vectors added or subtracted?

Vectors are added or subtracted by adding or subtracting their corresponding components. For example, to add two vectors A and B, you would add their x-components and their y-components separately to get the resulting vector's components.

4. What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Examples of scalars include temperature and mass, while examples of vectors include displacement and velocity.

5. How are vectors used in real-life applications?

Vectors are used in a wide range of real-life applications, including physics, engineering, and computer graphics. They can be used to model and analyze physical phenomena such as motion, forces, and electricity, and are also used in various computer programs and simulations.

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