Finding values to make a unit vector

In summary, the conversation is about finding all values of a such that the vector w=ai+\frac{a}{8}j is a unit vector. The solution is +/- \frac{8}{\sqrt{65}}, which was obtained by setting the magnitude of w equal to 1 and solving for a. However, there was an error in the attempt at a solution, as the equation 1=magw=\sqrt (a2+(a/8)2) was incorrectly simplified to 64=2a2 instead of 64=a2.
  • #1
Sammy600
2
0

Homework Statement


Find all values of a such that w=ai+[itex]\frac{a}{8}[/itex]j is a unit vector.

Homework Equations


unit vector has length of 1. and for a vector v unit vectors would be v/magv

The Attempt at a Solution


1=magw=\sqrt (a2+(a/8)2)
1=a2+(a2/64)
64=2a2
32=a2
a=[itex]\sqrt{32}[/itex]

i know that the solution is: +/- [itex]\frac{8}{\sqrt{65}}[/itex] but am at a loss as to how it was obtained. any help is appreciated.
 
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  • #2
sqrt(a^2+a^2/64)=1
so this means sqrt((64a^2+a^2)/64)=1
so sqrt(65a^2/64)=1
this means 65a^2/64=1
and then you get the result
 
  • #3
Sammy600 said:

Homework Statement


Find all values of a such that w=ai+[itex]\frac{a}{8}[/itex]j is a unit vector.

Homework Equations


unit vector has length of 1. and for a vector v unit vectors would be v/magv

The Attempt at a Solution


1=magw=\sqrt (a2+(a/8)2)
1=a2+(a2/64)
64=2a2
Your mistake is above. Multiply each term on the right side by 64. You don't get 2a2.
Sammy600 said:
32=a2
a=[itex]\sqrt{32}[/itex]

i know that the solution is: +/- [itex]\frac{8}{\sqrt{65}}[/itex] but am at a loss as to how it was obtained. any help is appreciated.
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is typically denoted by a hat (^) symbol above the vector's variable. It is used to represent direction without any consideration for length or scale.

2. Why is it important to find values to make a unit vector?

A unit vector is important because it helps to simplify and standardize calculations in vector operations. It also represents a direction without being affected by the length or scale of the vector, making it useful in various applications such as physics, engineering, and computer graphics.

3. How do you find values to make a unit vector?

To find values to make a unit vector, you first need to calculate the magnitude of the vector. Then, divide each component of the vector by its magnitude. This will result in a vector with a magnitude of 1, making it a unit vector.

4. Can any vector be turned into a unit vector?

Yes, any vector can be turned into a unit vector by following the steps mentioned above. It is important to note that the vector must have a non-zero magnitude in order for it to be converted into a unit vector.

5. What is the significance of unit vectors in physics?

In physics, unit vectors are used to represent direction in a three-dimensional coordinate system. They are essential for solving problems involving force, velocity, acceleration, and other vector quantities. Additionally, unit vectors allow for simplification of calculations and provide a standard way of representing direction in different physical systems.

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