Solving a Vector Problem: Length (17^1/2) and Same Direction as v = <7,0,-6>

In summary, the conversation discusses finding a vector that satisfies the conditions of having a length of 17^(1/2) and the same direction as the given vector v = <7,0,-6>. The individual attempts to solve this by finding the vector length and setting it equal to a multiple of v, but ultimately asks for clarification on their answer. Other individuals suggest finding the unit vector and using that to solve for the desired vector, and suggest that they check their answer with an instructor. The conversation ends with a suggestion that the book may have a slightly different expression for the answer.
  • #1
Kawrae
46
0
Find the vector that satisfies the stated conditions: Length (17^1/2) and same direction as v = <7,0,-6>.

The book gives the answer as being u = (1/5)<7,0,-6>. Having some trouble getting this answer... here is how I went about attempting to solve it:

1. Finding the vector length of the given vector.
||v|| = (7^2 + 0^2 + -6^2)^(1/2)
= (49+0+36)^(1/2)
= (85)^(1/2)

2. Setting the two vectors equal...
||u|| = (17)^(1/2) = k||v|| Since multiples of the vector will give same direction
(17)^(1/2)/||v|| = k

So k = (17)^(1/2)/(85)^(1/2)

So then my answer would be u = (17)^(1/2)/(85)^(1/2) <7,0,-6>.


Can anyone help point out where I am messing up?? Thank you
 
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  • #2
Work with vectors. Your first condition is u = k v. The second one is ||u||=||kv|| = 17^(1/2). This should work just fine.
 
  • #3
I'd recommend first finding the unit vector [tex]v = \frac{u}{||u||}[/tex], because you know that vector has a magnitude of 1. Then, you know that [tex]||kv||=\sqrt{17}[/tex], and you know that [tex]||v||=1[/tex], so [tex]k=\sqrt{17}[/tex]. So, your new vector [tex]w=kv=<\frac{7\sqrt{17}}{\sqrt{85}},0,\frac{-6\sqrt{17}}{\sqrt{85}}>=<7\sqrt{\frac{1}{5}},0,-6\sqrt{\frac{1}{5}}>[/tex] has a magnitude of [tex]\sqrt{17}[/tex]. So, your answer should be correct... I'd check the answer in your book with your instructor.
 
Last edited:
  • #4
Kawrae said:
Can anyone help point out where I am messing up?? Thank you

Are you sure that the book doesn't have the following?
[tex]\sqrt{\frac{1}{5}} \left<7,0-6\right>[/tex]

(Which is one of many possible expressions equivalent to the answer you've given.)
 

Related to Solving a Vector Problem: Length (17^1/2) and Same Direction as v = <7,0,-6>

1. What is a vector?

A vector is a mathematical quantity that has both magnitude (length) and direction. It is typically represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude.

2. How do you solve a vector problem?

To solve a vector problem, you first need to identify the given information, such as the magnitude and direction of the vector. Then, you can use mathematical operations, such as addition, subtraction, and scalar multiplication, to manipulate and solve the vector.

3. What is the length of a vector?

The length of a vector, also known as its magnitude, is the distance from the initial point to the terminal point of the vector. It is typically represented by the absolute value of the vector, denoted as ||v||.

4. What does it mean for two vectors to be in the same direction?

When two vectors are in the same direction, it means that they have the same orientation and are pointing in the same direction on a coordinate plane. This can be represented mathematically by comparing the unit vectors, which are vectors with a magnitude of 1, in the same direction.

5. How can you determine the direction of a vector?

The direction of a vector can be determined by using the unit vector in the same direction as the given vector. The unit vector is calculated by dividing the given vector by its magnitude, resulting in a vector with a magnitude of 1. The direction of the vector can then be described using angles or by using the direction cosines.

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